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WORKS OF PROF. L. E. DICKSON 





PUBLISHED BY 


JOHN WILEY 3k SONGS: 


Introduction to the Theory of Algebraic Equa= 
tions. 


Small 8vo, v + 104 pages. Cloth, $1.25 net. 
College Algebra. 


A text-book for colleges and technical schools, 


Small 8vo, vii+ 214 pages. Illustrated. Cloth, 
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Linear Groups with an Exposition of the Galois 
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8vo, x + 312 pages. Cloth, 12 marks. 





INTRODUCTION TO THE 


es Oe YO) 
ALGEBRAIC EQUATIONS. 


BY 


LEONARD EUGENE DICKSON, Pu.D., 


ASSISTANT PROFESSOR OF MATHEMATICS IN 
THE UNIVERSITY OF CHICAGO, 


ETS PT ae LON, 
FIRST THOUSAND. 


NEW YORK: 
JOHN WILEY & SONS. 
LonpoN: CHAPMAN & HALL, LIMITED. 
1903. 





oe 


aes Copyright, 1903, 
BY 


L. E. DICKSON. 





- ROBERT DRUMMOND, PRINTER, NEW YORK. 








rey Yd ae 


PREFACE. 


THE solution of the general quadratic equation was known as 
early as the ninth century; that of the general cubic and quartic 
equations was discovered in the sixteenth century. During the suc- 
ceeding two centuries many unsuccessful attempts were made to 
solve the general equations of the fifth and higher degrees. In 1770 


Lagrange analyzed the methods of his predecessors and traced all 


their results to one principle, that of rational resolvents, and proved 
that the general quintic equation cannot be solved by rational re- 
solvents. The impossibility of the algebraic solution of the general 
equation of degree n(n >4), whether by rational or irrational resoly- 


ents, was then proved by Abel, Wantzel, and Galois. Out of these 


algebraic investigations grew the theory of substitutions and groups. 
The first systematic study of substitutions was made by Cauchy 
(Journal de I’ école polytechnique, 1815). 

The subject is here presented in the historical order of its devel- 
opment. The First Part (pp. 1-41) is devoted to the Lagrange- 
Cauchy-Ahel theory of general algebraic equations. The Second 
Part (pp. 42-98) is devoted to Galois’ theory of algebraic equations, 
whether with arbitrary or special coefficients. The aim has been 
to make the presentation strictly elementary, with practically no 
dependence upon any branch of mathematics beyond elementary 
algebra. There occur numerous illustrative examples, as well as 
sets of elementary exercises. 

In the preparation of this book, the author has consulted, in 


addition to various articles in the journals, the following treatises: 
iii 


685414 


iv PREPAGE: 


Lagrange, Réflexions sur la résolution algébrique des équations; 
Jordan, T'raité des substitutions et des équations algébriques; Serret, 
Cours d’ Algéebre supérieure; Netto-Cole, T'heory of Substitutions and 
us Applications to Algebra; Weber, Lehrbuch der Algebra; Burn- 
side, The Theory of Groups Pierpont, Galois’ Theory of Algebrare 
Equations, Annals of Math., 2d ser., vols. 1 and 2; Bolza, On the 
Theory of Substitution-Groups and its Applications to Algebraic 
Equations, Amer. Journ. Math., vol. XIII. 

The author takes this opportunity to express his indebtedness 
to the following lecturers whose courses in group theory he has at- 
tended: Oscar Bolza in 1894, E. H. Moore in 1895, Sophus Lie in 
1896, Camille Jordan in 1897. 

But, of all the sources, the lectures and publications of Professor 
Bolza have been of the greatest aid to the author. In particular, 
the examples (§ 65) of the group of an equation have been borrowed 
with his permission from his lectures. 

The present elementary presentation of the theory is the out- 
come of lectures delivered by the author in 1897 at the University 
of California, in 1899 at the University of Texas, and twice in 1902 
at the University of Chicago. . 


CuicaGco, August, 1902 


TABLE OF CONTENTS. 


CHAPTER PAGES 


I. Solution of the General Quadratic, Cubic, and Quartic Equa- 
tions. Lagrange’s Theorem on the Irrationalities Entering 


RN ye as aie ba es oie o/c ldg's aw a oe « coats MiNemelT «8 1-9 

MENT 2 oe see Deiole sc o'h.<6 FG ee as 0.0 se aie oe ues 4 

II. Substitutions; Rational Functions...... abet s MME es 5 . 10-14 

| SME Feo cre he cacy iri aves aiding ce vila suites cngecae ss : 14 

Ht) Substitution Groups; Rational Functions..........ccececcs . 15-26 

Sau fre 5 LEP re Che 5 aio. in vee We 90 0/ne eels : 20 

IV. The General Equation from the Group Standpoint........... 27-41 

Ms fete ne Reig ce eae Os sess ont ghee 41 

WV eeearenraic Introduction to Galois’Theory........sccecssesece 42-47 

PeeeererroUp Ol an Piquation: ¢ ccc ccc cocewtescecesveseucs 48-63 

rer ae Sa eed pies ce vd acess ch tesneevevens 57-58 

VII. Solution by means of Resolvent Equations.............006. 64-72 

VIII. Regular Cyclic Equations; Abelian Equations.............. 73-78 

Seeeeriecrionstor Alecbraic Solvability..........ccccssccecvcces 79-86 

X. Metacyclic Equations; Galoisian Equations......... miata «cette 87-93 

XI. An Account of More Technical Results......... Rheate nee sles oe-OS 
APPENDIX. 

Pymamdecric Functions, .......2.esesee0e Mousseeocsescevec en Uo LUL 

On the General Equation ............. rive waves eecle seueew st Ulla 


INDEX... Tie caias cicidicisisietbia cons tcweeeeceeee coetveccuonlUs 
Vv 





THEORY OF ALGEBRAIC EQUATIONS. 


FIRST PART. 


THE LAGRANGE-ABEL-CAUCHY THEORY OF 
GENERAL ALGEBRAIC EQUATIONS. 





CHAPTER I. 


SOLUTION OF THE GENERAL QUADRATIC, CUBIC, AND QUARTIC 
EQUATIONS. LAGRANGE’S THEOREM* ON THE IRRATION- 
ALITIES ENTERING THE ROOTS. 


1. Quadratic equation. The roots of x?+pr+q=0 are 
2,=4(—ptV p?—4q), %,=4(—p—V p?—4q). 
By addition, subtraction, he multiplication, we get 
t+ 2, = Te Shes tama oe V p?—4q, Uyle= q. 
Hence the irrationality Vp?—4q, which occurs in the expressions 
for the roots, is rationally expressible in terms of the roots, being 
equal to x,—a,. Unlike the last function, the functions x,+2, 
and 2,7, are symmetric in the roots and are rational functions of 


the coefficients. 
2. Cubic equation. The general cubic equation may be written 


(1) x? —c,x?+¢,%—¢c,=0. 
Setting c=y+4c,, the equation (1) takes the simpler form 
(2) y*+ pyt+q=9, 


* Reflexions sur la résolution algébrique des équations, Giuvres de Lagrange, 
Paris, 1869, vol. 3; first printed by the Berlin Academy, 1770-71. 


GENERAL QUADRATIC, CUBIC, AND QUARTIC. [Cu. I 


if we make use of the abbreviations 

(3) P=O—$C", Q=—C,+$C,0,—¥yC,*. 

The cubic (2), lacking the square of the unknown quantity, is 
called the reduced cubic equation. When it is solved, the roots 
of (1) are found by the relation x=y+ 4¢,. 

The cubic (2) was first solved by Scipio Ferreo before 1505. 
The solution was rediscovered by Tartaglia and imparted to * 
Cardan under promises of secrecy. But Cardan broke his promises 
and published the rules in 1545 in his Ars Magna, so that the 
formule bear the name of Cardan. The following method of 
deriving them is essentially that given by Hudde in 1650. By 
the transformation 





(4) y=2—, 

the cubic (2) becomes 23— Ae +q=0, whence ~ 
p? 

(5) 2+ gz? — 5 =0. 


Solving the latter as a quadratic equation for 23, we get 
2=—hgtVR, R=1¢?+ayp. 
Denote a definite one of the cube roots of —4qg+V R by 
/ —MtVR. 
The other two cube roots are then 
WY -WtVR, 0 ¥/—3g+VR, 


where w is an imaginary cube root of unity found as follows. The 
three cube roots of unity are the roots of the equation 


re—1=0, or (r—1)(r?+r+1)=0. 
The roots of r?+r+1=0 are —$+4V —3=o and —4—4V —3=@?, 
Then 
(6) w?+w+1=0, =k, 


s 








Src. 2] THEORY OF ALGEBRAIC EQUATIONS. 3 


In view of the relation 
(—49+V R)(—3g-V R) =4@?— R= — dp’, 





a particular cube root Aye, R may be chosen so that 
aig VR. e/ q-V R= - 3 
w 2/3 + VR - 0 &/'—4qg-V R= — 4p, 
w? Y/ At VR w8/—3q—-VR = — 4p. 
Hence the six roots of equation (5) may be separated into pairs 


in such a way that the product of two in any pair is —3$p. The 


root paired with z is therefore =2, and their sum z= is, in. 


view of (4), a root y of the cubic (2). In particular, the two roots 
of a pair lead to the same value of y, so that the siz roots of (5) 
lead to only three roots of the cubic, thereby explaining an apparent 
difficulty. Since the sum of the two roots of any pair of roots 
of (5) leads to a root of the cubic (2), we obtain Cardan’s formuls 
for the roots y,, Yy, yz of (2): 


y= —Ht+VR+4Y/ -hq-VF, 
(7) yaw 8/ —4g+V R+w? 8&/ —3q—V R, 
ts) — hgtVR+w 2/ — 4q—V R. 
Multiplying these expressions by 1, w?, w and adding, we get, 
by (6), ie 
Y -MAV RABY Foy toys)= Xo 
Using the multipliers 1, w, w?, we get, similarly, 
Y= hq-VR=4Y, + oy Foy). 
Cubing these two expressions and subtracting the results, we get 
VR= Fe { (ys + o7Y2+ wy)? — (Yr t oY, +0"ys)*} 


=¥—3y, wu wo Ys—Y3)) 





4 GENERAL QUADRATIC, CUBIC, AND QUARTIC. [Cu I 


upon applying the Factor Theorem and the identity w—w?=V —3. 

Hence all the irrationalities occurring in the roots (7) are rationally 

expressible in terms of the roots, a result first shown by Lagrange. 
The function 


(Y1— Yo) (Y2— Ys) (Ys— V1)" = — 21 — 4p 
is called the discriminant of the cubic (2). 
The roots of the general cubic (1) are 


Lr=Yz+SCQy, Le=Yot Cy, Te=Y3+ Hey. 
ve Ly Le=Yr— Yo, Ly —XLE=Y2— Ya, %3—M%=Y3s—- 1; 
(8) (XL — X_)(L_—Xg)(X3—2%,) = (Y, —Y-)(Yo— Ys) Yea a) 


= VV R= 3 VP 


EXERCISES. 


1. Show that 2, +077, + 02,=Yy,;+ 07y,+ wys, ©, + W2,+ W723 =Y; + OY, + W7Ys. ~ 

2. The cubic (2) has one real root and two imaginary roots if R>0O; three 
real roots, two of which are equal, if R=0; three real and distinct roots if 
R<0O (the so-called irreducible case). 

3. Show that the discriminant (x,—2,)?(x,—23)?(a3—2,)”. of the cubic (1) 
equals 

C,2c,? + 18¢,c.¢, —4c,3 —4c, °c, —27¢,7. 
Hint: Use formula (8) in connection with (3). 


4. Show that the nine expressions 3/ =a Ee R +2/ Shoe R, where 
all combinations of the cube roots are taken, are the roots of the cubics 
yitpytq=0, y'twpyt+q=0, y*+w*pytq=0. 


~5. Show that y,t+y.t+y3=0, YYotYYstyYs=P, YWYYs=—4- 
‘6 Show that 7,142.42, =¢,, 0,%,+2,T%+2p%3=Cy, 1%o%, =, US 
How may these results be derived directly from equation (1)? 


3. Aside from the factor 4, the roots of the sextic (5) are ~ 


$,=%,+wr,+0*Xs, Py=X,+ Wt, + way, | 
(r= =X, +0t,+W'X,, $s=0°~=T,+W%,+0"X,, 
$3= WP, =U + OX, +W°X,, Jo=Wby=X,+ Wk, + 07s. 


These functions differ only in the permutations of 2%,, 2, x. As 
there are just six permutations of three letters, these functions 


Src. 3] THEORY OF ALGEBRAIC EQUATIONS. 5 


give all that can be obtained from ¢, by permuting 2,, 2,, x. For 
this reason, ¢, is called a six-valuwed function. 
Lagrange’s @ priori solution of the general cubic (1) consists 


in determining these six functions ¢,,...,¢, directly. They are 
the roots of the sextic equation (t—¢,)...(t—d¢,)=0, whose 
coefficients are symmetric functions of ¢,,..., ¢, and consequently 


symmetric functions of x,, 7,7, and hence * are rationally expressible 
in terms of ¢,, G, ¢,. Since ¢,=wd,, d,=wy,, etc., we have by (6) 


(t —,)(t —Po)(t <i s) =F — d,°, 
(t mr d,)(t a bs)(t vr Je) = — p,°. 


Hence the resolvent sextic becomes 
(9) Yo — (PF +h,/)P+¢,° be=0. 
But Py Py= 2" + Uy? +H? + (w+ w?) (1,2 + 14%, +225) 
= (1, +2, + %5)?—3( 1,2, + 1,23 + 1X3) = C,?—3e, 
in view of Ex. 6, page 4. Also, ¢,°+¢,3 equals 


AG? 1° + 2,°) — 3( 2,70, + 2,257 1,72, + 2,Xq7 + 1y7Ly + Loy”) + 1277, 2oc 
= 3(42+2,°+ 275°) —(€, +2, +2%)°+ 18x,2,2, 
= 2c,°—9¢,c, + 27¢s. 


Hence equation (9) becomes 
t® — (2c,3 —9c,c, + 27c,)t? + (c,?—3c,)3=0. 


Solving it as a quadratic equation for t°, we obtain two roots 0 
and 6’, and then obtain 


b=V0, hav. 
Here “6 may be chosen to be an arbitrary one of the cube roots 
of 0, but 0’ is then that definite cube root of 0’ for which 
(10) VO 0 =c,2—3ey. 
We have therefore the following known expressions: 
t,twr,+w2x,=V 0, t,+w2,+twt,=V0, 2%, +2,+2,=¢. 


* The fundamental theorem on symmetric functions is proved in the 
Appendix. 


6 GENERAL QUADRATIC, CUBIC, AND QUARTIC. — [Cu. 1 


Multiplying them by 1, 1,1; then by w?, w, 1; and finally by w, w?, 1; 
and adding the resulting equations in each case, we get 
t=1etVOtvV6’), 
(11) t,=4(ce,+wV 0+ 8/6’), 
ts=4(c, tw VO+a? 8/6’). 
4, Quartic equation. The general equation of degree four, 
(ee) ei +ax>+br?+cr+d=0, 
may be written in the form 
(a? + 40x)? = (4a?—b)a?—cx—d. 
With Ferrari, we add (x?+4ax)y+4y? to each member. Then 
(13) (x? +4ax-+4y)?=(40?—b+y)x? + (Fay—c)a+hy’—d. 


We seek a value y, of y such that the second member of (13) shall 
be a perfect square. Set 


(14) a’—4b+4y,=#?. 
The condition for a perfect square requires that 


als = 2 
(5) H+ Gay, oe tay —d= (Je EZ)” 


ay tie zay,—C€\* __ (ay— 6)” 
Be t a?—4b+4y," 
Hence y, must be a root of the cubic, called the resolvent, 
(16) y®— by? + (ac—4d)y—a?d+ 4bd—c?=0. 


In view of (15), equation (13) leads to the two quadratic 
equations 


(17) x’? + (ga—3t)x + dy, — (Say, —c)/t=0, 

(18) w+ (Fat Ht)x + by, + (Gay, —©)/t=0. 
Let x, and x, be the roots of (17), x, and x, the roots of (18). Then 
LZ +X,= —Zatht, Xx,=sy,— (gay, —¢)/t, 
%g+2,=—ta—ht, 2x,0,=4y,+ (4ay,—0)/t. 


i 


Src. 5] THEORY OF ALGEBRAIC EQUATIONS. 7 


By addition and subtraction, we get 
(19) y+ X,—XLz—Uy=b, My, + UzL~= Yj. 


In solving (17) and (18), two radicals are introduced, one equal 
to a,—2, and the other equal to z,—2, (see § 1). Hence all the 
irrationalities entering the expressions for the roots of the general 
quartic are rational functions of its roots. 

If, instead of y,, another root of the resolvent cubic (16) be 
employed, quadratic equations different from (17) and (18) are 
obtained, such, however, that their four roots are 2,, 1, 2, %,; 
but paired differently. It is therefore natural to expect that the 
three roots of (16) are 


(20) Yr=Uy%_ Ls q,  Yg=UyX3 + XX, Yg=XyXy~ + Loz. 

It is shown in the next section that this inference is correct. 

5. Without having recourse to Ferrari’s device, the two quad- 
ratic equations whose roots are the four roots of the general quartic 
equation (12) may be obtained by an @ priori study of the rational 
functions 2,%,+%,%, and x#,+%,—2,—2,=t. The three quantities 
(20) are the roots of (y—y,)(y—Yy2)(y—Yys3) =9, or 


C21) Yt Yet Ys)Ye + (YYot Vist YoY s)Y — YrYo's= 9. 
Its coefficients may be expressed * as rational functions of a,b,c, d: 


Yr t Yot Yg= Lye, + U3ly+ LL + Ll, +X,L,+2,X,=), 
MYT YrYst YoYg= — 40X20 324 
+ (Uy +2 q +3 + 4) (X,2Ly + LLC, + U,X3L, + L052) 
=ac—Ad, 
YYoYg= (L,Lyly + L,LyCy +L, LzL, + LyL_L,)" 
+204 § (Ly + Ly +%y +24)? —4(0,H+0,%_+...4+U50,)} 
=¢?+d(a?—4b), 


* This is due to the fact (shown in § 29, Ex. 2, and § 30) that any per- 
mutation of x,, %,, %, x, merely permutes ¥;, Y, 3, So that any symmetric 
function of y,, y., y; is asymmetric function of 2, x, x3, x, and hence rationally 
expressible in terms of a, , c. d. 


8 GENERAL QUADRATIC, CUBIC, AND QUARTIC. [Cu. I 


Hence equation (21) is identical with the resolvent (16). Next, 


t? = (4, +2, +23 + 4)? — A(X, + %q)(%g +24) 
= 0? —A(a,0,+0,%,+ ... +20) +42,2,+475%, 
=a’?—4b+4y,. 


Again, 2,+2,+%7,+%2,=—a. Hence 
t,+%=t(t—a), %+2,=3(—t—a). 


To find x,x, and x,7,, we note that their sum is y,, while 


—t—a t—a 
—C=2,%,(%,+2,) +2,0,(2,+2,) =7,2, (==) tog, ) , 


., U4%,=(c—fay,+ 4ty,)/t, T0,=(—e+ Zay,+ $ty,)/t. 


Hence 2, and 2, are the roots of (17), x, and x, are the roots of (18). 

6. Lagrange’s a priorz solution of the quartic (12) is quite 
similar to the preceding. A root y,=2,7,+2,x, of the cubic (16) 
is first obtained. Then 2,7,=2, and x,7,=z, are the roots of 


2?7—y z2+d=0. 
Then x,+2, and 73+, are found from the relations 


(%,+%,) + (X%3+2,) = —a, 





2o(@, +22) +2, (ag + 2) = 2g Ly + Ugh Co FL Xots eg 
—az,+e az,—C 
'. YTn= eae 
&— &, &—&, 


‘Hence x, and 2, are given by a quadratic, as also x, and a,. 
7. In solving the auxiliary cubic (16), the first irrationality 
entering (see § 2) is 
A= (Y,— Yo) Yo— Ys) (Yi — Ys)- 
But Ya — Yo= (LZ —%4)(,— Ls), 
Yo—Yg= (4%, —X_)(Xz— Xs), Yr —Yg= (11 — Lp) (Lp — A), 
in view of (20). Hence 


(22) 4= (x, — L)(X, <4 Ty) (Ly or 4) (Lp ie Lg) (Xp —%,)(%3—2%,). 


SEC. 7] THEORY OF ALGEBRAIC EQUATIONS. 
By § 2, the reduced form of (16) is 7*+Py7+Q=0, where 


ee eA ely? 
(23) 1A ac—4d—4)?, 


Q=—a’d+ gabce+ $bd—c?— °,b%. 


Applying (8), with a change of sign, we get 





(24) 4=6V —3ViQ?+s,P%. 


CHAPTER II. 
SUBSTITUTIONS ; RATIONAL FUNCTIONS. 


8. The operation which replaces 2, by ra, %, by Xp, 13 by 2;,..-, 
t, by t,, where a, &,..., vy form a permutation of 1) 2;..eeee 
is called a substitution on 2,,.%, %,,..., Yn. It is usually des- 


ignated 
Hct eedey MS hake 
Ta Uy Ly .-. Lyf] 


But the order of the columns is immaterial; the substitution may 
also be written 


To ie emer or (tn M1 2 Xs +e 
Ly La ty «+. Ly)’ Ly La Ly ty ...J?""” 


The substitution which leaves every letter unaltered, 


is called the identical substitution and is designated I. 

9. THrorEM. The number of distinct substitutions on n letters 
is ni=n(n—1)...3-2-1. 

For, to every permutation of the n letters there corresponds a 
substitution. 


ExAmpuLeE. The 3!=6 substitutions on n=3 letters are: 
XT Ly Xz UT Ly 3 XU X_ Xs 
T= » a 4 hs b= , ’ 
X, Uz Lz U Xz X U3 XX 
Tin taate d Die Ley tg ae ae 
C= ; —- F c= . 
Ly a Ee 6 Py. A fo Ce 


Applying these substitutions to the function Y=2, + wt, +wx;, we obtain 
the following six distinct functions (cf. § 3); 


$1=2,+0%,+0'r,=$, $a=X,+ 0x, +72, =0%%, $b =X3+ wr, + wx, =v, 
2 = , 2 ye = , 
Po=X,+ wr, + WX, Pa =%_+ WX, +W'X,=wW pe, He=X,+ Hr, + wr, =e, 
Io 


SEC. 10] THEORY OF ALGEBRAIC EQUATIONS. It 


Applying them to the function 6=(2x,—2,)(«,—2;)(x,—2,), we obtain 
$1 = fa= fo = 9, hc= ha = be = — . 
Hence ¢ remains unaltered by J, a, b, but is changed by e, d, e. 


10. Product. Apply first a substitution s and afterwards a 
substitution t, where 


ae a eh ae ge ss. =. 
La yg... Me La Ly... Ly! 
The resulting permutation 2,’, Uy, + ++ 5 Ly’ CaN be obtained directly 
from the original permutation «,, 2,..., 2» by applying a single 
substitution, namely, 
um (2 Feed ee). 
Lat Ler «2. By! 


We say that wu is the product of s by ¢ and write w=st. 

Similarly, stv denotes the substitution w which arises by apply- 
ing first s, then ¢, and finally v, so that stu=uv=w. The order 
of applying the factors is from left to right.* 

Examp.es. For the substitutions on three letters (§ 9), 

ab=ba=I, ac=d;. ca=e, ad=e, da=e, 
aa=b, bb=a, abec=Ic=c, aca=da=c. 

Applying the substitution a to the function ¢, we get ¢a; applying the 
substitution c to ¢a, we get ¢a. Hence ¢ac=¢a. Likewise ¢an=¢1=¢, 
goa=¢. ; 

11. Multiplication of substitutions is not commutative in 
general. | 

Thus, in the preceding example, ac¥ca, ad¥da. But ab=ba, 
so that a and 0 are said to be commutative. 

12. Multiplication of substitutions is associative: st-v=s-tv. 

Let s, t, and their product st=wu have the notations of § 10. If 


by AG We ee ad Ee H by H 03 er acs 
pee. 8 Pa) ea Wet eo = (a2 O78 Sp 
Ha!’ af! eee Cyt! Xa 7 ee Ly 
Ss (2 --- Bn) _ (2, Ty ++. Fn) (Ta... \ ony 
2 ” ’ ‘ ye ” © 
Lau Ln" Sad ay, La tp Raccoon Wg averse 


EXAMPLE. For 3 letters, ac-a=da=c, a-ca=ae=c. 











“* This is the modern use. The inverse order ts, vis was used by Cayley 
and Serret. 


[2 SUBSTITUTIONS ; RATIONAL FUNCTIONS. (Cu. II 


13. Powers. We write s? for ss, s* for sss, ete. Then 
(25) Sgr = gmat (m and n positive integers). 

For, by the associative law, s™s"=s™.sst 1=smtign t= | | 

14. Period. Since there is only a finite number n! of distinct 
substitutions on 7 letters, some of the powers 

Siybtn 8 wee eee 

must be equal, say s”=s™*", where m and n are positive integers. 
Then s™=s™s", in view of (25). Hence s” leaves unaltered each 
of the n letters, so that s"=T. 

The least positive integer o such that s’ =I is called the period 
of s. It follows that 


(26) 88 nies ee 


are all distinct; while s°t!, s°t?,..., s%*—!, 5% are repetitions 
of the substitutions (26). Hence the first o powers are repeated 
periodically in the infinite series of powers. 


ExampLes. From the example in § 10, we get 


a?=b, a’=a’a=ba=I, whence ais of period 3; 
b?=a, b3=b’%b>=ab=I, whence bis of period 3; 
c, d, e are of period 2; I is of period 1. 


15. Inverse substitution. To every substitution s there corre- 
sponds one and only one substitution s’ such that ss’=J. If 


3He . sy then sf=(%* fd ‘ 


La Up eee Ly aX, Xs ere Xn 


Evidently s’s=J. We call s’ the inverse of s and denote it hence- 
forun bys 9... tence 


SSan== 5 Sarde mn) es: 


If s is of period o, then s~'=s’—*'. Since s replaces a rational 
function f=/(%,,...,%n) by fs=/(®a,...,%v), 8~* replaces f, by f. 


EXxAMPLEs. For the substitutions on 3 letters (§ 9), 
a fA mt [La Ms ep) ns has Ve 
— (a v3 Hy) ster & U2 =) = vy a ai? 


oa PE =te™ naa OMe pe ye se fe 
bvleag, ori=¢, «dtd. foes anes [oe 


Src. 16] THEORY OF ALGEBRAIC EQUATIONS. 13 


These results also follow from those of the examples in § 14. For the 
functions of § 9 the substitution a replaces ¢ by ¢a; a~!=b replaces ga by ¢. 


16. THErorEM. Jf st=sr, then t=r. 
Multiplying st and sr on the left by s~1, we get 


Gest fn San Si 


17. TororeM. If ts=rs, then t=r. 
18. Abbreviated notation for substitutions. Substitutions like 


ess & Lo ah b= & Ls A), q= & Lz Xy ay 
a Pre 418 Tel rea 
which replace the first letter in the upper row by the second letter 
in the upper row, the second by the third letter in the upper row, 
and so on, finally, the last letter-of the upper row by the first letter 
of the upper row, are called circular substitutions or cycles. In- 


stead of the earlier double-row notation, we employ a single-row 
notation for cycles. Thus 


A=(X4Xp_X_), O=(XyzX4X), T= (L.T.7,T,). 


Evidently (2,225) = (Xo1'3%,) =(X%,2,), Since each replaces x, by 
X, L, by 3, and x, by x,. A cycle rs not altered by a cyclic permu- 
tation of its letters. 

Any substitution can be expressed as a product of circular 
substitutions affecting different letters. Thus 


Tie ©, Hy Va Ey Ls, Le\ _ 
ie L, :) = (%,)(2223), © Tg Le LyX, “) (©,Lg%5) (Log) (L,). 
A eycle of a single letter is usually suppressed, with the under- 
standing that a letter not expressed is unaltered by the substitution. 
Thus (2,)(a%3) 18 written (7,2). 
A circular substitution of two letters is called a transposition. 
19. Tables of all substitutions on n letters, for n=3, 4, 5. 
For n=3, the 3!=6 substitutions are (compare § 9): 
T=identity, a=(2,7,7,), b=(x,2,2,), 
C= (x23), A=(X4%,), e=(X2,). 


14 SUBSTITUTIONS ; RATIONAL FUNCTIONS. [Cu. Il 


For n=4, the 24 substitutions are (only the indices being written): 
I =identity; 
6 transpositions: (12), (13), (14), (23), (24), (34); 
8 cycles of 3 letters: (123), (132), (124), (142), (134), (1438), (234), 
(243) ; 
6 cycles of 4 letters: (1234), (1248), (1324), (1342), (1423), (1432) ; 
3 products of 2 transpositions: (12)(34), (13)(24), (14)(23). 


For n=5 the 5!=120 substitutions include 
I =identity; 


5-4 
oe. =10 transpositions of type (12); 


os =20 cycles of type (123); 


oes"? 30 cycles of type (1234) ; 


pee = 24 cycles of type (12345); 
5-3=15 * products of type (12)(84); 
20 + products of type (123)(45). 


EXERCISES. 


1. The period of (123...) isn; its inverse is (nn—1...321). 

2. The period of any substitution is the least common multiple of the 
periods of its cycles. Thus (123)(45) is of period 6. 

3. Give the number of substitutions on 6 letters of each type. 

4. Show that the function 2x,7,+2;x, is unaltered by the substitutions J, 

(2X), (1 3Xq), (Ly%2)(T3Xu), (XyXy) (yy), (1 pXq) (Loy), (TpLs% yy), (Ly yys). 

5. Show that x, +32, is changed into 7,23 -+2,0, by (ps), (0,4), (%4%:%), 
(2,1 2X4), (XX Xy), (Lely), (Ly~yL4Ls), (1X32 4X2). 

6. Write down the eight substitutions on four letters not given in Exs, 
4 and 5, and show that each changes 7,2,+2 3%, Into 1,%,+2% 03. 


* Since the omitted letter may be any one of five, while one of the four 
chosen letters may be associated with any one of the other three letters. 
+ The same number as of type (123), since (45) =(54). 


CHAPTER III. 


SUBSTITUTION GROUPS; RATIONAL FUNCTIONS. 


20. A set of distinct substitutions s,, s,,..., Sm forms a group 
if the product of any two of them (whether equal or different) is a 
substitution of the set. The number m of distinct substitutions 
in a group jis called its order, the number n of letters operated on 
by its substitutions is called its degree. The group is designated 
Ge: 

All the n! substitutions on n letters form a group, called the 
symmetric group on n letters G”. In fact, the product of any 
two substitutions on n letters is a substitution on » letters. The 
name of this group is derived from the fact that its substitutions 
leave unaltered any rational symmetric function of the letters. 


ExAmpLeE 1. For the six substitutions on n=3 letters, given in § 9, the 
multiplication table is as follows: * 


Logon a. & 

fs Lee. Ue" ay e 

a Fy i gee Aa Oy: Ree » 

ese b els oi e,7d ¢ 
Cc Gi Oia. Did 

d de cee ened Les Db 

e ee Cape tiie.) 


Thus ad =e is given in the intersection of row a and column d. 
EXAMPLE 2. The substitutions J, a, b form a group with the multiplica- 
tion table 


to ent 

4 7. i herrea te 
a ai a 

b ee a! 





* Tt was partially established in the example of § 10. 
15 


16 SUBSTITUTION GROUPS; RATIONAL FUNCTIONS. [Cu. Il 


If s is a substitution of period m, the substitutions 
LTS See See 
form a group of order m called a cyclic group. 


EXAMPLE 3. J, a=(123), b=a?=(132) form a cyclic group (Ex. 2). 
Examp.e 4, I, s=(123)(45), s?=(132), s?=(45), s*=(123), s°=(132)(45) 
form a cyclic group of order 6 and degree 5. 


21. FUNDAMENTAL THEOREM. All the substitutions on ,, 
Yo, +++, Xn which leave unaltered a rational function P(x,, Lo, .-- , Ln) 
form a group G. 

Let $, denote the function obtained by applying to ¢ the sub- 
stitution s. Ifaand b are two substitutions which leave ¢ unaltered, 
then ¢.=¢%, ¢s=9-. Hence 


(pa)o=(P)o= Go=; or pab=9P- 
Hence the product ab is one of the substitutions which leave 
unaltered. Hence the set has the group property. 
The group G is called the group of the function ¢, while ¢ is 
said to belong to the group G. 


ExaAMPLeE 1. The only substitutions on 3 letters which leave unaltered 
the function (x,—2») (% —23)(%;—2,) are (by § 9) I, a=(a,%2%,), b=(x,7,2,). 
Hence they form a group (compare Ex. 2, § 20). Another function belonging 
to this group is 


(a, +wx,+w72,)?, wan imaginary cube root of unity. 


EXAMPLE 2. The only substitution on 3 letters which leaves unaltered 
x, +wx,+w*x, is the identity J ($9). Thus the substitution J alone forms 
a group G, of order 1. 

EXxampLe 3. The rational functions occurring in the solution of the 
quartic equation (§ 4) furnish the following substitution groups on four 
letters: 

a) The symmetric group G,, of all the substitutions on 4 letters. 

b) The group to which the function y, =2,2, + 23x, belongs (Exs. 4-6, p. 14): 


Gs={I, (12), (34), (12)(34), (13)(24), (14)(23), (1324), (1428)}. 
c) Since y,=2,%,+2,%, is derived from y,=27,2%7,+2%3%, by interchanging 
x, and xs, the group of y, is derived from G,; by interchanging x, and x; within 
its substitutions. Hence the group of y, is 


Gs’ = (1, (18), (24), (18)(24), (12)(84), (14)(32), (1284), (1432) }. 


SEc. 22] THEORY OF ALGEBRAIC EQUATIONS. 17 


d) The group of y,;=2,%,+2,%,, derived from G, by interchanging z, 
and &,, 1S: 


G,” = (J, (14), (32), (14)(32), (13)(42), (12)(43), (1842), (1243) }. 

e) The function x,+2,—2x,—2, belongs to the group 

Since all the substitutions of H, are contained in the group Gy, H, is called 
a subgroup of G,. But H, is not a subgroup of G,’. 

jf) The function ¢=y,+wy,+w7y, or 

PH XyL, + U3lq + O(LyX3 + LyX) + w(L,X 4+ Lys), 

remains unaltered by the substitutions which leave y,, y,, and y, simulta- 
neously unaltered and by no other substitutions. Hence the group of ¢ is 


composed of the substitutions common to the three groups G;, G,’, G,”’, 
forming their greatest common subgroup: 


G,={I, r=(12)(34), s =(13) (24), ¢=(14)(23)} 
That these four substitutions form a group may be verified directly: 
phe og? rate taf 
rs=sr=t, rt=tr=s, st=ts=r. 


Hence anv two of its substitutions are commutative. This commutative 
group G, is therefore a subgroup of Gs, G,’, and G,”’. 


22. THEOREM. Livery substitution can be expressed as a product 
of transpositions in various ways. 

Any substitution can be expressed as a product of cycles on 
different letters (§ 18). A single cycle on n letters can be expressed 
as a product of n—1 transpositions: 


(1234... n)=(12)(13)(14) .. . (In). 


Exampurs. ~(123)(456) =(12)(13)(45)(46), 
(132) =(13)(12) =(12)(23) =(12)(23)(45)(45). 


23. THroREM. Oj the various decompositions of a given substi- 
tution s into a product of transpositions, all contain an even number 
of transpositions (whence s is called an even substitution), or all 
contain an odd number of transpositions (whence s 1s called an od 
substitution). 


18 SUBSTITUTION GROUPS; RATIONAL FUNCTIONS. {Cu. UI 


A single transposition changes the sign of the alternating 
function * 


p=(x,—2X_)(@, —Xg)(%, — 2X4) see (1,—2n) 
slp rahe) (tte 2) came (X,—2Xn) 
e . e e . Gan Besa 


Thus (,2,) affects only the terms in the first and second lines of 
the product, and replaces them by 


(1 —2X,)(L_— Xz) (,— Ly) . . . (Ly— Ap) 
(1, —%3)(2,— 4%)... (Hae 
Hence, if s is the product of an even number of transpositions, 
it leaves ¢ unaltered; if s is the product of an odd number of trans- 
positions, it changes ¢ into —¢. 
CoroLuaRyY. The totality of even substitutions on 7 letters ° 
forms a group, called the alternating group on n letters. 
EXAMPLE 1. The alternating group on 3 letters is ($§ 9, 19) 
G3) = tis (123), (132) }. 
EXAMPLE 2. The alternating group on 4 letters is (§ 19) 
Gy,(4) = {I, (12)(34), (13)(24), (14)(23), and the 8 cycles of three letters}. 
24. THrorEeM. The order of the alternating group on n letters 
is 4-n! 
Denote the distinct even substitutions by 
(e) C1) Co, Czy 2 + +» Cke- 
Let ¢ be a transposition. Then the products 
(0) Biber Cal, Went ts tay Cee 


are all distinct (§ 17) and being odd are all different from the 
substitutions (e). Moreover, every odd substitution s occurs in 





* It may be expressed as the determinant 





MA A eS a 
2 —1 

Le ety ees Saekee 

e . e * . . aa 

Ltn fen ee ie ee 





Src. 25] THEORY OF ALGEBRAIC EQUATIONS. 19 


the set (0), since st is even and hence identical with a certain e;, 
so that 

Se Cian 0,1. 
Hence the 2k substitutions given by (e) and (0) furnish all the n! 
substitutions on n letters without repetitions. Hence k=}-n! 

20. As shown in § 21, every rational function d(a7,,..., Xn) 
belongs to a certain group G of substitutions on 2,,..., Z,, namely, 
is unaltered by the substitutions of G and changed by all other 
substitutions on 2,,...,2%,. We next prove the inverse theorem: 


Given a group G of substitutions on x,,...,%n, we can construct 
a rational function f(2,,...,Xn) belonging to G. 
Let G= {a=I, b,c,..., 1} and consider the function 
V=M,2,4+ Mot, + ... +MnXn, 
where m,, m,,..., M» are all distinct. Then V is an n!-valued 
function. Applying to V the substitutions of G, we get 
(27) eS | el eee ae ay 


all of which are distinct. Applying to (27) any substitution c 
of G, we get 
(28) Wren eV Le 


These values are a pertiutation of the values (27). since ac, bc,... , Ic 
all belong to the group G and are all distinct (§ 17). Hence any 
symmetric function of V,, Vy,..., V; 1s unaltered by all the 
substitutions of G. By suitable choice of the parameter p, the 
. symmetric function 
p=(0—V)(o—Ve)(o—V-) --. (e—Vi) 
will be altered by every substitution s not in G. Indeed, 
p= (p— Vie)(o— Vis)(o— Vicelenet (o— Ve} 
is not identical with ¢ since V, is different from V, Vs, V.,...+ Vi. 
ExamplLeE 1. For G={I, a= (14,25), b=(a,2,7,)}, take 
V =2,+07,+ 0%. 
Then Va=w?V, Vp=wV. Hence 
VtVatVo=(1+0+0)V=0, VVa+VVs+VaVo=0, VVaVo=V". 
The function V* belongs to G (see Ex. 1, § 21). - 


20 SUBSTITUTION GROUPS ; RATIONAL FUNCTIONS. (Cu. II 


EXAMPLE 2. For G={I, c=(a,%,)}, take the V of Ex.1. Then 
VVc=(4, +02, + 7x,)(%,+w2,+72,) =c,?—3c, 
is unaltered by all six substitutions on the three letters. But 
b=(e—V)(o—Ve) =p? — (2%, — 2, —23)9 + Cy? —3e, 


for 00, is changed by every substitution on the letters not in G. Hence, 
for any 0+0, ¢ belongs to G. 


EXERCISES 
Ex. 1. If wisa primitive uth root of unity, 
(%, +02, +07%3+ ... +wh—tey)4 


belongs to the cyclic group {J, a, a?,..., a*—'}, where a=(a, 4%... 2p). 

Ex. 2. Taking V =2,+7%,—z,—1a, and s=(x,%.)(%,%,), show that 
VV s=i(x,—2,)?+i(2,—2,)? belongs to G, of § 21, that V+Ve belongs to H, 
of § 21, while (o—V)(o—Vs), for 040, belongs to theegroup {J, s} 

Ex 3. Taking V=a,+7ix,—2x,—ix, and t=(2,23)(2,2,), show that VV; — 
belongs to the group {J, ¢} 

Ex. 4. If a,, a,. ., an are any distinct numbers, the function 


V =2,%1 X,%2 - Ln in 


is n!-valued, and V+ V,+Vc+ ...+V: belongs to {I,b,c,. .,l}. 
Ex. 5. If é belongs to G and ¢’ belongs to G’, constants a and a’ exist 
such that af+a’¢’ belongs to the greatest common subgroup of G and G’, 


26. THroreM. The order of a subgroup ts a divisor of the order 
c) the group. 

Consider a group G of order N and a subgroup H composed of 
the substitutions 
(29) heels dudes tise. 


If G contains no further substitutions. N=P, and the theorem 
is true. Let next G contain a substitution g, not in H. Then 
G contains the products 


(30) Gas RaGos MeGay-- any LEQo- 


The latter are all distinct (§ 17), and all different from the sub- 
stitutions (29), since hag,=hg requires that g,=ha'hg=a sub- 


Sec. 27] THEORY OF ALGEBRAIC EQUATIONS. 21 


stitution of H contrary to hypothesis. Hence the substitutions 
(29) and (80) give 2P distinct substitutions of G. If there are 
no other substitutions in G, N=2P and the theorem is true. Let 
next G contain a substitution g, not in one of the sets (29) and (30). 
Then G contains 


(31) Js) RoGa: heJs,- - - » APY; 


As before, the substitutions (31) are all distinct and all different 
from the substitutions (29). Moreover, they are all different from 
the substitutions (30), since hag,=hsg, requires that g,=hi'heg, 
shall belong to the set (30), contrary to hypothesis. We now 
have 3P distinct substitutions of G. Either N=3P or else G 
contains a substitution g, not in one of the sets (29), (80). (31). 
In the latter case, G contains the products 


(32) Oi Wels eae amd 


all of which are distinct and all different from the substitutions 
(29), (30), (31), so that we have 4P distinct substitutions. Pro- 
ceeding in this way, we finally reach a last set of P substitutions 


(33) Jv, h.gv, hey; RTO} hpq., 
since the order of H is finite (§ 9). Hence N=vP. 


DEFINITION. The number =>, is called the index of G 
Y 


the subgroup H under G, and the relation is exhibited in I 
the adjacent scheme. 

CoroLuaRy. The order of any group H of substitutions on n 
letters is a divisor of n! Indeed H is a subgroup of the symmetric 
group G,, on n letters. 

27. THrorremM. The period of any substitution contained in a 
group G of order N is a divisor of N. 

If the group @ contains a substitution s of period P, it contains 
the cyclic subgroup H of order P: 


iteet Bestar Spr kt Slot 


Then, by § 26, P is a divisor of N. 


22 SUBSTITUTION GROUPS ; RATIONAL FUNCTIONS. [Cu. Il 


CoroLLARyY.* If the order N of a group G is a prime number, 
G is a cyclic group composed of the first N powers of a substitution — 
of period N. 

28. As shown in § 26, the N substitutions of a group G@ can 
be arranged in a rectangular array with the substitutions of any 
subgroup # in the first row: 

Wee Palace Wicks, aa ee 
Jo Mog. Nyy .-- hrge 
Js hogs ggg --- Pgs 


gv hogy hego .-- beg r 


Here g,=/, 9, 93,+++, 9» are called the right-hand multipliers. 
They may be chosen in various ways: g, is any substitution of G 
not in the first row; g, any substitution of G not in the first and 
second rows; g, any substitution of G not in the first, second, and 
third rows; ete. 

Similarly, a rectangular array for the substitutions of G may 
be formed by employing left-hand multipliers. 

29. THrorEM. Ij ¢ is a rational function of x,...., Ln belonging 
to a subgroup H of index v under G, then ¢ is v-valued under G. 

Apply to ¢ all the N substitutions of G arranged in a rect- 
angular array, as in § 28. All the substitutions belonging to a ~ 
row give the same value since 


Prida - (Dn, aoe as (P),, =. Pons 


Hence there result at most v values. But, if 


Yo.= a, (B<oye 


then oo, —% so that g.gg' is a substitution h; leaving ¢ 
* This result is a special case of the following theorems, proved in any 
treatise on groups: 

If the order of a group is divisible by a prime number p, the group contains — 
a subgroup of order p (Cauchy) 

If pt is the highest power of the prime number p dividing the order of a 
group, the group contains a subgroup of order p* (Sylow). 


Src. 30] THEORY OF ALGEBRAIC EQUATIONS. 23 


unaltered. Hence g.=h,g,, contrary to the assumption made in 
forming the rectangular array. 

Derinition. The v distinct functions ¢, ¢,,, ¢,,.--,Yy, are 
called the conjugate values of / under the group G. 

Taking G to be the symmetric group G,,, we obtain Lagrange’s 
result: 

The number of distinct values which a rational function of n 
letters takes when operated on by all n! substitutions vs a divisor of n! 


EXAMPLE 1, To find the distinct conjugate values of the functions 
A =(X,—2X,)(X,—2Xs)(X¥z—2,),  O=(X, +x, + we)? 


under the symmetric group G;, on 3 letters, we note that they belong to the 
subgroup G;={J, a=(2,2,7,), b=(x,2,7,)}, as remarked in § 21, Ex. 1. The 
rectangular array and the conjugate values are: : 


6 


I: A=(X,2,23), b=(X,25%2) 4 
Gc 


C=(X,%2), AC=(X32,), bc =(2x,%,) —A 








EXAMPLE 2. To obtain the conjugate values of 2,7,+2,7, under the 
symmetric group G,, on 4 letters, we rearrange the results of Exs. 4, 5, 6, 
page 14, and exhibit a rectangular array of the substitutions of G,, with 
those of G; in the first row: 


tf (12), (34), (12)(34), (13)(24), (14)(23), (1824), (1428) | 7,7,+4%32, 
(234), (1342), (23), (132), (143), (124), (14), (1243) |2,2,+2,7, 
(243), (1432), (24), (142), (123), (134), (1234), (13), \ LyX, + LLy 





380. THrorEM. The p distinct values which a rational function 
P(%,..., Ln) takes when operated on by all n! substitutions are the 
roots of an equation of degree o whose coefficients are rational functions 
of the elementary symmetric functions 


eee tet tat... en, CoH 2,T, +2, 0,4... 4+EnWn; i+, 
Cy = LX, . . ° Ln. 


Let the e distinct values of A(x,,..., Xn) be designated 


(35) i=, gy $y ---) Pp 
They are the roots of an equation (y—¢,)(y—¢,) ... (y—d¢p) =0 
whose coefficients ¢,+@,+..-.+p...., Edidy ... Gp are symmetric 


functions of ¢,, @.,..., dp. After proving that they are symmetric 


24 SUBSTITUTION GROUPS; RATIONAL FUNCTIONS. {Cu. Il 


functions of x,, %,..., &n, we may conclude (Appendix) that they 
are rational functions of the expressions (34). We have therefore 
only to prove that any substitution s on 72,, ..., %, merely 
interchanges the functions (35). Let s replace the functions 
(35) by respectively 


(36) Pir Pos Par + +s Poe 

In the first place, each ¢’ is identical with a function (35). 
For, there exists a substitution ¢ which replaces ¢, by ¢;, and s 
replaces $; by ¢;, so that ts replaces ¢, by ¢j. Hence there is a 
substitution on 2,,...,%, which replaces ¢, by ¢;, so that ¢} 
occurs in the set (35). 

In the second place, the functions (86) are all distinct. For, 
if d=}, we obtain, upon applying the substitution s~', d:=¢d; 
contrary to assumption. 

DEFINITION. The equation having the roots (35) is called the 
resolvent equation for ¢@. 

Compare the solution of the general cubic (§ 3) and general quartic (§ 5). 

31. LAGRANGE’s THEOREM. Ij a rational function f(X,, La,..-) Ln) 
remains unaltered by all the substitutions which leave another rational 
function P(a,, Lo,..., Ln) unaltered, then f rs a rational function 
OF ONGC ess 2" 4 Ore 

The function ¢ belongs to a certain group 


H=jh=J, wee ode. 


Let v be the index of H under the symmetric group G,. Consider 
a rectangular array of the substitutions of G,; with those of H 
in the first row: 


| hs as P=; 
J» hogy - ee Py, =r Pin= Pe 


Dy a oo. Apgy Ogre iy, Pg, =Pr 


Then ¢,, /o,..., dy are all distinct (§ 29); but 6, d.,..., ¢» need 
not be distinct since ¢ belongs to a group G which may be larger 
than H. Under any substitution s on 2,, %,..., %», the functions 


Src. 31] THEORY OF ALGEBRAIC EQUATIONS. 25 


d,, do ...,%, are merely permuted (§ 30). Moreover, if s replaces 
_ g; by gy, it replaces ¢; by ¢; Set 
g(t) =(t¢—¢,)(¢—¢,) .. . (¢—d»), 


ima ( 724 +5 +... +25), 


so that A(t) is an integral function of degree »v—1 in ¢. Since 
A(t) remains unaltered under every substitution s, its coefficients 
are rational symmetric functions of 2,, %,..., %, and hence are 
rational functions of the expressions (34). Taking ¢,=/¢ for ¢, 
we get * 


Mp.) == (Pi — $2) (Pi- Fg)» «(G1 - Pv) "Px =9' (Ps) Py 











iP) 
37) =i 
= g (f) 
The theorem may be given the convenient symbolic form: 
G:¢ 
wee then p—Rat. Func. (d; c,,... 5 en). 
H :¢ 


Taking first H=G and next H=J, we obtain the corollaries: 

CorouuarRyY 1. [} two rational functions belong to the same group, 
either 1s a rational function of the other and ¢,, ¢,,.. +, €n- 

Ceroruany 2. Lrery rational function of 2,, t%,..., Ln UG 
rational function of any n!-valued function (such as V of § 25) and 
Clases Cn 

EXAMPLE 1. The functions 4 and 0 of Ex. 1, § 29, belong to the same 
group a. We may therefore express 4 in terms of 0. By $§.2, 3, 


(e*— —3c, 3C))? 
ae 


B\/ — 3 4 =(2,4 wry w2,)?— (2, + wr, + w23)3 = —0. 


The expression for 0=¢,’ in terms of 4 is given in § 34 below. 





* The relation (37) is vahd as long as 2, 2%,..., Zn denote indeterminate 
quantities, since ¢,,...,¢» are algebraically distinct so that g’(%) is not 
identically zero. In case special values are assigned to 2,,..., %» such that 
two or more of the functions ¢,,..., ¢, become numerically equal, then 
~ g'(¢) =0, and ¢is not a rational function of ¢, c¢,,..., cn. In this case,-see 
yee. Giuvres, vol. 3, pp. 374-388; Serret, Algébre, Il, pp. 434-441. 
But this subject is considered i in Part II. 


26 SUBSTITUTION GROUPS; RATIONAL FUNCTIONS {[Cu. IIl 


EXAMPLE 2, The function y,=2,7,+2,;7, belongs to the group G, and 
t=27,+27,—x,—2, belongs to the subgroup H, (§ 21). Hence y, is a rational 
function of ¢ and the coefficients a, b, c, d of the equation whose roots are 
Ly, Ly, Lz, X. By $5, y,=7(t?—a’ +40). 

EXAMPLE 3. The function ¢,=2,+wz,+w’x; has 3!=6 values. Hence 
every rational function of 2, x, x, is a rational function of ¢, and ¢, C2, C3. 
The expressions for 2, 2, 2; themselves follow from the formule (11) of § 3. 
Thus 


c,?—3¢, 
t1=3 (c.+4.+ 5) . 


G:¢d 
32. THEOREM. Jf vy | _ , then ~ satisfics an equation of degree v 
1d 
whose coefficients are rational junctions of , ¢,..., Cn: 
As in § 29, we consider the v conjugate values of ¢ under G: 


g, $9, Yo.) Oa $o,- 


Under any substitution of the group G, these values are merely 
permuted amongst themselves. Hence any symmetric function 


of them is unaltered under every substitution of G and therefore, 


by Lagrange’s Theorem, is a rational function of ¢, ¢,..., Cy. 
The same is therefore true of the coefficients of the equation 


(w—$)(w—gy,) ... (W—Py,) =. 





CHAPTER IV. 
THE GENERAL EQUATION FROM THE GROUP STANDPOINT. 


33. In the light of the preceding theorems, we now reconsider 
Cardan’s solution (§ 2) of the reduced cubie equation y?+ py+q=0. 
The determination of its roots y,, Y¥, yz depends upon the chain 
of resolvent equations: | 


2p a 
Ga T4+5-, where §=——"(y—2)(Yo—Ys) (Yo) 


= S46, where 2=3(y,+wy.+w7Ys) } 


Se Rae OE 2) 
Ye a5? Yo = We 27? Y3 >We 27° 


Initially given are the elementary symmetric functions 
Yt YotYs=9, YYotYVYstYYs=P, —YWYYs=4%, 
belonging to the symmetric group G, on Y,, Yo, Y3- Solving a 
quadratic resolvent equation, we find the two-valued function &, 
which belongs to the subgroup G, of G, (§ 21, Ex. 1). Solving 
next a cubic resolvent equation, we find the six-valued function 2, 
which belongs to the subgroup G, of G, (§ 21, Ex. 2). Then y,, y, ys 
are rational functions of z, p, g, since they belong to the respective 
groups 
Ge’ = (1, (Yos)}, Go’ = tL, (Vis) $s Ge!” = 11, (Wye) $5 
each containing G, (also direct from § 381, Cor. 2). From the 
group standpoint, the solution is therefore expressed by the scheme: 


Ge: DP, q 


2 | 
Ag ee 7 porns Gel Yg 
G22 Ga:2 Gita Giz 

27 


28 GENERAL EQUATION FROM THE GROUP STANDPOINT. [Cu. IV 


34, The same method leads to a solution of the general cubic 
x? —c,x? +¢,7 —¢,=0. 
To the symmetric group G, on 2,, 2, 7, belong the functions 
Lz +Xo+Le=Cy, LiX_etXzXyetLX,—=Cy, U,HoXy=Ceg. 
To the subgroup G,= {J, (2,224), (2,%%)} belongs the function 
A= (X,—X2)(X_— Xg)(X3— 2). 
In view of Ex. 3, page 4, 4 is a root of the binomial resolvent 
4? =¢,7c,” + 18¢, CC, — 4,3 — 4¢,3¢, — 27c,?. 
By § 3 and § 2, we have for ¢,=2,+w2,+w2,. $4=2,+wt,+ws, 
dg it+t¢ f= 2¢7%—9c,c,+27c,, 
by pie BV —3(2, — Hq) (L_— Lg) (Lg = 2) = — 8V—3 4. 
*, b 8=4(2c,3—9c,c,+27¢,—3V —3 4), 
$F=4(2c,3—9c,c,+27¢,+3V —3 4). 
After determining * ¢, by extracting a cube root, the value of 
dy is (§ 3) 
, Jy = (C1? —3¢,) + gy. 
Then, as in § 3, 2, 7, x, are rationally expressible in terms of ¢,: 
BH=Uaqthtys), 2=3(4, tod, +w9,), %=4F(¢,+wg,+wg,). 
35. The solution given in § 5 of the general quartic equation 
(12) x*+ax?+ bx? +cx+d=0 
may be exhibited from the group standpoint by the scheme: 


CPR OR! 
| 


ae 2% 2 
fy SY, = 2X, +0, V(t, +a, 7, ap 


TT 36, Beda We ee 


Ue 


Here H,= {I, (,2,)}, H,’={I, (a,2,)}, G, and H, being given in § 21. 








* For another method see Ex. 4, page 41. 


Sxc. 36] THEORY OF ALGEBRAIC EQUATIONS. 20 


36. Lagrange’s second solution of (12) is based upon the direct 
computation of the function 2,+2,—2x,—.,. Its six conjugate 
values under G,, are +¢,, +¢,, +t, where 

Q=X,t%,—My—-T, = X%,+%3—X,—X%, t3=X,+X,—X_,—y. 
The resolvent sextic is therefore 

(2? —1,?)(7? —1,7)( 7? —#,7) =0. 
Its coefficients may be computed * easily by observing that 
t’?=a?—4b+4y,, t,2=a°—4b+4y,, t,?=a?—4b+4y,, 
as follows from § 5. Using the results there established, we get 
ty? +t,’+ t,? = 3a? —12b+4 4(y,+ y,4+ y,) =3a?—8b, 
t,7t,?+ t,7t,’ + t,t,” = 3(a —4b)?+ 8(a? —4b)(y,+ y+ Yo) 


+ 16(Y:Yo+ YiYat Yos) 
= 3a*— 16a7b+ 16b?+ 16ac— 64d, 


t,t,7t,” = (a? Beto) + 4(a? z 4b)*(y, T Yet Ys) 
+ 16(a* — 4b) (YYot YiYst Yes) + 649, YoYs 
= {8c+ a(a?—4b)}?. 

The resolvent becomes a cubic equation upon setting t?=o. De- 
note its roots by o,=1,’, o,=t,’, o,=t,’. Then 

L4+ t,—X,—2,=V 6, £,+2,—21,—2,=V 0, 

Uy X, —%,—2z=V/G,, 2+ L,+ t+ 2,= —4. 
From these we get 


m= —at+Vo,4Vo,4Vo,), =H —a+Vo,—Va,-Va,), 
t= 4( —A—W/G,4-V/o,—Vo,), 2,=1(—a—V0,—Va,+ V9) 


The signs of V/ o, and Vo, may be chosen arbitrarily, while tha‘ 


«| 


of V 0, follows from 
(39) V6iV6,V 6, = ttt, = 4ab —8e—a’. 
Indeed, we may determine the sign in 

t,t,t,—= + {8c+ a(a’—4b)} 


* Compare Ex. 5, page 41. 


30 GENERAL EQUATION FROM THE GROUP STANDPOINT. [Cu. IV 


by taking z,=1, 7,=27,=2,=0, whence a= —1, b=c=d=0, t,t,t,=1. 

37. The following solution of the quartic is of greater interest 
as it leads directly to a 24-valued furiction V, in terms of which 
all the roots are expressed rationally. As in § 5, we determine 
y, and t, belonging to G, and H, respectively, by solving a cubie 
and a quadratic equation. To the subgroup 


G= (1, (Xy%,)(_,) } 
of H, belongs the function ¢=V?, while to G, belongs V, where 
V=(2,—2,)+ ua, —2,)- 


Under H,, ¢ takes a second value ¢,={(2,—2%,)—(7,—2,)}?. 


Then 
z’—($+ ¢,)e+ $f,=0 

is the resolvent equation for ¢. But 

a= (x —2,)?+ (@—2,)"}? = {a? — 2b -2y, }?= 4480? 8b, 
Ut (y= 24) a, = er) 2 = 2(%,—2%,+ L,—X,)(@, —Ty eee 

= 2(4ab—8c— a?) +t, 
in view of (39). After finding ¢~ and ¢,, we get 
VaV¢. Vi=Vb,=(a,—2,) -(a,—2,); 

(40) V,=4(3a?—8b—?)+V. 
Having the four functions t, V, V,, and 7,+.2,+2%,+%,=—a, we get 


m=H—att+V+V,), 2%=+(—a+t-V-V)), 


Ot) med =a tt) ee 


38. The solution of the general cubic (§ 34) and the solution of 
the general quartic (§ 37) each consists essentially in finding the 
value of a function which is altered by every substitution on the 
roots and which therefore belongs to the identity group G,. Like- 
wise, the general equation of degree n, 


(42) 2 OTN Coe ta one om LG 


could be completely solved if we could determine one value ofa 
function belonging to the group G,; for example, 


Src. 38] THEORY ©OF “ALGEBRAIC EQUATIONS. 31 


(43) V=mM,2,tMytot+ ... +MnXn (m’s all distinct). 


In fact, each x; 1s a rational function of V,c¢,,...,¢,n by § 31. For 
the cubic and quartic, the scheme for determining such a function 
V was as follows: 


Pires, vol, Cy Go. Grand, 
2 | 3 | 
G,: (2%, +wx,+w?2,)% Gil Filet Pel, 
3 | 2 | 
G,:2,twt,+ wr, A 4:%,+%,—4,—%, 
9 


Gy: (2, — 1p +-12%,—12,)? 
2 | 
G,:%,—X_ +14, —12, 
The same plan of solution applied to (42) gives the following scheme: 


Pitas Cay eee 7on 


d | 
(Sede ered C eds Cae a teas te =O) 
ue | 
‘Sep elitty sons Ca) TC otis sO 
M:¢ 
a | 
Cave VOI Oe Cit.) Ga i... =O. 


Such resolvent equations would exist in view of the theorem of 
§ 32. In case the resolvent equations were all binomial, the 
function V (and hence g,,..., X,,.) would be found by the extraction 
of roots of known quantities, so that the equation would be solvable 
by radicals. We may limit the discussion to binomial equations 
of prime degree, since z?¢=A may be replaced by the chain of 
equations z?=u, w4=A. The following question therefore arises: 


G: 
If vy |  , when will the resolvent equation for / take the form 
i: 


34 


(44) gv=Rat. Func. (¢,¢,- «5 Cp). 


32, GENERAL EQUATION FROM THE GROUP STANDPOINT. [Cu. IV 


Since v is assumed to be prime, there exists a primitive vth root 
of unity, namely a number w having the properties 


w’=1, wk] for any positive integer k< v. 


Hence the roots of (44) may be written 


(45) CeO, Us ere same 

Let ¢,=¥¢, ¢2,..., Yy denote the conjugate functions to ¢ under G 
(their number is v by § 29). Now ¢ belongs to the group H by 
hypothesis. Let ¢, belong to the group H,, ¢, to Hs, ..., g, to Hy. 


Since the roots (45) differ only by constant factors, they belong 
to the same group. Hence a necessary condition is that 


Ho eee 


39. The first problem is to determine the group to which belongs 
the function ¢, into which ¢ is changed by a substitution s, when 
it 1s given that ¢ belongs to the group - 


HM eth=lo Io ee : 
If a substitution o leaves ¢, unaltered, so that ¢,,= qs, then 
ie oes pee Pss-t = i. 
Hence sos~!=h, where h is a substitution of H. Then 
o=s" hs. 
Inversely, every substitution s~hs leaves ds unalterel. Hence 
ds belongs to the group 
187 has =], 8 saligs 2 eee ee aoe 
which will be designated s~'Hs. We may state the theorem: 
If ¢ belongs to the subgroup H of index v under G, the conjugates 


¢, $955 ae a) Jar 
of } under G, belong to the respective groups _ 
Tp SOse rg gag kOe. LL Oae 
DeFrinitions. ‘The latter groups are said to form a set of con- 


jugate subgroups of G. In ease they are all identical, H is called 
a self-conjugate subgroup of G (or an invariant subgroup of G). 


Src. 40] THEORY OF ALGEBRAIC EQUATIONS. 33 


Hence a necessary condition that the general equation of 
degree n shall be solvable by radicals under the plan of solution 
proposed in § 38 is that each group in the series shall be a self- 
conjugate subgroup of prime index under the preceding group. 

Note that the group G,={/} is self-conjugate under every 
group G since g-'Ig=I. 

EXAMPLE 1. Let G be the symmetric group G, on 3 letters and let H 
be the group G,;={J, (2,273), (a@,%2,)}. Let g,=(x,7;). Then 

$ =(2,+0r,+*2,)?, $o, = (2+ wx, + w2;)* 
form a set of conjugate functions under G. Now ¢ belongs to H and ¢, 
belongs to the group {J, (a,7,2,), (@,2,7,)}, whose substitutions are derived 
from those of H by interchanging the letters x, and 2,, since that interchange 
replaces ¢ by ¢g,. To proceed by the general method, we would compute 
(20s) (L223) (Lys) =(HyLsCp),  (W_Xy) —'(XyL3Xp) (L_T,) = (TT). 
By either method we find that the group of ¢ and ¢g, are identical, so that 


G, is self-conjugate under G, Also, G, is self-conjugate under G,. Hence 
the necessary condition that the general cubic shall be solvable by radicals 
is satisfied 

EXAMPLE 2. Consider the conjugate values 2, 2, x, of x, under G,: 


i (223) | 
Jz =(X,%q), (2X3) Jo = (242,75) | Te 


93=(2y25),  (X_X2) 93 =(XyX2%y) | Lz 
Hence H = {J, (x,7;)} is not self-conjugate under G,. Here 


9, 'Hg, = (I, (a ,23) } ~H, 93 'Hg3= if, (x,22) ; “HH, 


40. Derinitions. Two substitutions a and a’ of a group G 
are called conjugate under ( if there exists a substitution g belong- 
ing to G such that g~'ag=a’. Then a’ is called the transform 
of a by g. 

There is a simple method of finding g~'ag without performing 
the actual multiplication. Suppose first that a is a circular sub- 
stitution, say a=(af70), while g is any substitution, say 


34 GENERAL EQUATION FROM THE GROUP STANDPOINT. [Cu. 1V 


Hence g~'!ag=(a’f’y’0’) may be obtained by applying the substi- 
tution g to the letters of the cycle a=(afy0). 

Let next a=a,a,a,..., Where a,, d,... are circular substitu- 
tions. Then . 


g—ag=g~'a,9-9~'dog-G "dg... 


Hence g~ag is obtained by applying g within the eyeles of a. 
Thus (123)-".(12)(34) (123) =(23)(14). 
CoROLLARY. Since any substitution transforms an even sub- 
stitution into an even substitution, the alternating group Gj, 1s a 
self-conjugate subgroup of the symmetric group Gy). 


41. Turorem. Of the following groups on four letters: 


Gy, Gy, Gy= 11, (12)(34), (13)(24), (14)(23)}, 
6,= {TDG Gea 


each is a self-conjugate subgroup oj the preceding group. 

By the Corollary of $ 40, G;,, is self-conjugate under G.,. To 
show that G, is self-conjugate under G,, (as well as under G,,), 
we observe that G, contains all the substitutions of the type (a@f)(70), 
while the latter is transformed into a substitution of the form 
(a’3’)(7'9") by any given substitution on four letters. That G, 
is self-conjugate under G, follows from the fact that (12)(34), 
(13)(24), (14)(23) all transform (12)(34) into itself.* 


42. The necessary condition ($39) that the general quartic 
x'+anr*+ ba*+cx+d=0 


shall be solvable by radicals is satisfied in view of the preceding 
theorem. We proceed to determine a chain of binomial resolvent 
equations of prime degree which leads to a 24-valued function 


V=2,—2,4+1%,—12,, 








* This also follows from § 21, Ex. (/), since rs=sr gives s—'rs =r. 





Suc. 42] THEORY OF ALGEBRAIC EQUATIONS. 35 


in terms of which the roots 2,, X., 3, %, are rationally expressible. 
Let 
(20) YAU Lo A Lely, Yo=HUXyLgt Uy, Yg= ot hg, 
as in $4. The scheme for the solution is the following: 
Pri axon c,d 
Gy d =(X,—22)(H,—Xg)(@y — 4) (Lp — Vy) (Lp — Ly) (Lz — 4) 
ae 1 Dy=Y, FWY. + wy; 


Ae 2A = G+ (4, +2,—23—2,) 
2 
G,:V =2,—2,+1%,—12, 


Referring to formule (22), (23), (24) of § 7, and setting P= —4J, 
Q=16/, we get 
4=16V P—27J?, 
er ead alo 8 
he tel ot ri Ge 1G 216; 
Hence 4 is a root of the binomial resolvent 4?=256(/*—27J’). 
The resolvent for ¢, is the binomial equation 


(p—9$1)(P—wg,)($—-07h,) =P? — 9° =0. 


By Lagrange’s Theorem, ¢,° is a rational function of 4, a, b, c, d. 
To determine this function, set d,=Yy, +w?Y,+ wy. Then ($§ 2, 7) 


$3—$2=38V —3(Y,—Y2)(Yo—Ys) (Ys Ys) = —8V —3 4 
fet by =2(y2+Yy2+Ys°) + L2y,YoY3+3(w+w")d, 
where O=Y,"Yo+YYo + YY: +YYs +Y2°Yst+YoYs° Satisfies the rela- 
tions 
(Yr tYot Ys) (YiYo+ YY s+ Yols) = 9+ 38Y:YoYss 
(Y; + Yo+ Ys)? = 30 + BY, YoYg t+ Yy>+ Yn? + Y;°. 
. Po FG F=2(y, + Yt Ys)>— WY + Yot Ys(YiYot YiYst Yos) + 27y,YoYs 
= 2b*— 9b(ac—4d) +27(c?+a2d—4bd) = — 432 


36 GENERAL EQUATION FROM THE GROUP STANDPOINT. [Cu. IV 
upon applying the relations in § 5. Hence 
Jb 2=413V —34—216J. 


In view of Lagrange’s Theorem, y,, y,, and y, are rational functions 
of ¢, These functions may be determined as follows: 


ONS a che Yo Ya. +(w +w?)(Y,Yo + Y1Y3 + YxYz) 
=(Y,+Y2+ Ys)? — 3(Y1Y2+ YrYs + YoYs) 
=6*—3ac+12d=H. 


H 
From Y%t+YotYys=b, YyHWY.t+w7ys=hy, Yy+w7Y,+wYs= = 





et 
n= 4(b+$.47), w= a(d-+0%g.+), n= 4(d+op. +). 


Setting t=2,+7,—2,—2,, we obtain for A=¢,/t the binomial 
resolvent 
P=,’ + (a? —46+4y,), 


upon replacing ?? by its value given in § 5. Next, we have (§ 37) 
V?= (1-2)? — (&g— 24)? + 20(4 — Lp) (Ly — Ly) 


4ab—S8c—a? 
= eae oe e + 22(4fo— Ys) 


= -(4ab—80—a") +3V3($,-7). 


The values of 2, 2, Ys, 2, are then given by (41) in connection 
with (40). 


SERIES OF COMPOSITION OF THE SYMMETRIC GROUP ON n LETTERS. 


43. Drrinitions. Let a given group G have a maximal self- 
conjugate subgroup H, namely, a self-conjugate subgroup of G 
which is not contained in a larger self-conjugate subgroup of G. 
Let H have a maximal self-conjugate subgroup K. Such a series 
of groups, terminating with the identity group G,, 


G) He Ken, ee 


Src. 44] THEORY OF ALGEBRAIC EQUATIONS. Rf 


in which each group is a maximal self-conjugate subgroup of the 
preceding group, forms a series of composition of G. The num- 
bers A (the index of H under G), » (the index of A under H),..., 
o (the index of G, under M) are called the factors of composition 
of G. 

If the series is composed of the groups G@ and G, alone, the 
group G is called simple. ‘Thus a simple group is one containing 
no self-conjugate subgroup other than itself and the identity group. 
A group which is not simple is called a composite group. 

ExampteE 1. Fo she symmetric group on 3 letters, a series of composition 
is G,, G3, G, (see Ex. 1, § 39). Since the indices 2, 3 are prime numbers, the 
self-conjugate subgroups are maximal (see § 26). 

EXAMPLE 2. A series of composition of the symmetric group on 4 letters 


is Gy, Gy, G,, G., G, (§ 41), the indices being prime numbers. 
EXAMPLE 3. A cyclic group of prime order is a simple group (§ 26). 


44, Lemma. Jf a group on n letters contains all circular sub- 
stitutions on 3 of the n letters, vt is ether the symmetric group Gn, 
or else the alternating group Gy). 

It is required to show that every even substitution s can be 
expressed as a product of circular substitutions on 3 letters. Let 


s=hl, eS a boy —sloy 


where ¢,,...,¢, are transpositions (§$§ 22, 23), and 4,42. If 4 
and t, have one letter in common, then 


tity = (a8) (ay) =(a67). 


_If, however, ¢, and ¢, have no letter in common, then 


tyt,= (a3)(70) = (a9) (ar)(7a)(79) = (487) (a0). 


Similarly, i,t, is either the identity or else equivalent to one cycle 
on 3 letters or to a product of two such cycles. 
Hence the group contains all even substitutions on the n letters. 
45. THrorem. The symmetric group on n>A4 letters contains 
no selj-conjugate subgroup besides itself, the identity G,, and the 
alternating group Gin, so that the latter is the only maximal self- 
conjugate subgroup of Gy) (n>A4). 


38 GENERAL EQUATION FROM THE GROUP STANDPOINT. |Cu. IV 


That the alternating group is self-conJugate under the symmetric 
_ group was shown in § 40. 

Let G,, have a self-conjugate Se eee H which contains a 
substitution s not the identity J. 

Suppose first that s contains cycles of more than 2 letters: 


s=(aheU aa) jc eee 


Let a, 8, 0 be any three of the n letters and 7, «,..., o,... the 
remaining n—3 letters. Then H contains the substitutions 


= (aby... dEb..)..., = Gar... ES...) .05, 


the letters indicated by dots in s, being the same as the correspond- 
ing letters in s,. The fact that s, (and likewise s,) belongs to H 


follows since 
pa (tOc... def... 
Na Poy). 23 BO one 


is a substitution on the n letters which transforms s into s, (§ 40), 
while any substitution o of G,, transforms a substitution s of the 
selj-conjugate subgroup H into a substitution belonging to H 
(§ 39). Since H is a group, it contains the product s,s,~!, which 
reduces to (afd). Hence H contains a circular substitution on 
3 letters chosen arbitrarily from the n letters. Hence ZH is either 
Gn Or Gyn) (§ 44). 

Suppose next that s contains only transpositions and at least 
two transpositions. The case s=(ab)(ac)...=(abc)... has been 
treated. Let therefore 


s=(ab)(cd)(ef) .. . (Im). 


Let a, 2, 7, 0 be any four of the n letters, and ¢, ¢d,..., A, w the 
others. ‘hen the self-conjugate subgroup H contains the sub- 
stitutions 


=(aB)(70)(eP) ..- CA), 8:= (ar) (G0) (ed) «. . (Aw) 
and therefore also the product s,s,~', which reduces to (ad)(By). 


Src. 46] THEORY OF ALGEBRAIC EQUATIONS. 39 


Since n>4, there is a letter o different from a, 2, 7, 0. Hence H 
contains (a)($7) and therefore the product 
(ad) (87) -(ae)(G7) = (ade). 

It follows as before that H is either Gy; or Gyn). 

Suppcse finally that s=(ab). Then the self-conjugate subgroup 
H contains evcry transposition, so that H=G,,.. 

46. THrorem. The alternating group on n>4 letters is simple. 

Let G:,: have a self-conjugate subgroup H larger than the 
identity group G,. Of the substitutions of H different from the 
identical substitution /, consider those which affect the least 
number of letters. All the cycles of any one of them must contain 
the same number cf letters; otherwise a suitable power would 
affect fewer letters withcut reducing to the identity J. Again, 
none of these substitutions contains more than 3 letters in any 
cycle. Tor, if H contains 

pers ee ayn) 
then H contains its transform by the even substitution o= (234): 
ieee So atv 0) ie). oe; 
where the dots indicate the same letters as in s. Hence H would 
contain 
gs tel a2) 
affecting fewer letters than does s. Finally, none of the substi- 
tutions in question contain more than a single cycle. For, if H 
- contains either ¢ or s, where 
bat oA). , S—(123)(456)......, 

it would contain the transform of one of them by the even substi- 
tution x=(125) and consequently either ¢-«~'tx or s7!-x7'1sx. 
The latter leaves 4 unaltered and affects no letter not contained 
in s; the former leaves 3 and 4 unaltered and affects but a single 
letter 5 not contained in ¢t. In either case, there would be a reduc- 
tion in the number of letters affected. 

The substitutions, different from J, which affect the least num- 
ber of letters are therefore of one cf the types (ab), (abc). The 
former is excluded as it is odd. Hence H contains a substitution 


40 GENERAL EQUATION FROM THE GROUP STANDPOINT. [Cu. IV 


(abc). Let a, 8, 7 be any three of the n letters, 0, ¢,..., v the 
others. Then (abc) is transformed into (afr) by erther of the 
substitutions 

al ae oe 8) s= (0 eee 

GO BaD See ye af yoo pears 

where the dots in 7 indicate the same letters asin s. Since r=s(de), 
one of the substitutions 7, s is even and hence in Gy. Hence, 
for n>4, H contains all the circular substitutions on 3 of the n 
letters, so that H=Gj,). 

47. It follows from the two preceding theorems that, for n>4, 
there 1s a single series of composition of the symmetric group on n 
letters: Gn:, Gin}, G,. The theorem holds also for n=8, since 
the only subgroup of G, of order 3 is G,, while the three subgroups 
of G, of order 2 are not self-conjugate (§ 39, Ex. 2). The case 
n=4 is exceptional, since G,, contains the self-conjugate subgroup 
G, (§ 41). 

Except for n=4, the factors of composition of the symmetric group 
on n letters are 2 and 3n!. 

48. It was proposed in § 38 to solve the general equation of 
degree n by means of a chain of binomial resolvent equations of 
prime degrees such that a root of each is expressible as a rational 
function of the roots 2,, %,..., %, of that general equation. As 
shown in §§ 38-389, a necessary condition is the existence of a 
series of groups 


(46) Gin, aK ee eines 


each a self-conjugate subgroup of prime index under the preceding 
group. In the language of § 43, this condition requires that Gp; 
shall have a series of composition (46) with the factors of com- 
position all prime. By § 47, this condition is not satisfied if n}5, 
since 4n!is then not prime. But the condition is satisfied if n=3 
or if n=4 (§ 39, Ex. 1; § 41). Under the proposed plan cf solu- 
tion, the general equation of degree n>4 is therefore not solvable 
by radicals, whereas the general cubic and general quartic equa- 
tions are solvable by radicals under this plan (§ 34, § 42). 


Src. 48] THEORY OF ALGEBRAIC EQUATIONS. 41 


To complete the proof of the impossibility of the solution by 
radicals of the general equation of degree n>4, it remains to show 
that the proposed plan is the only possible method. This* was 
done by Abel (@uwvres, vol. 1, page 66) in 1826 by means of the 
theorem : 

Every equation which rs solvable by radicals can be reduced to a 
chain of binomial equations of prime degrees whose roots are rational 
functions of the roots of the given equation. 

As the direct proof of this proposition from our present stand- 
point is quite lengthy, it will be deferred to Part II (see § 94), 
where a proof is given in connection with the more general theory 
due to Galois. 


EXERCISES. 

1. If H={I, h,... , hp} is a subgroup of G of index 2, H is self-conjugate 
under G. 

Hint: The substitutions of G not in H may be written g, gh,,..., ghp} 
or also g, hyg,..., hpg. Hence every hgg is some gha, so that for every ha, 
g—‘hgg is some ha. 

2. The group G, of § 21 has the self-conjugate subgroups G,, G,, H,, 
C,={I, (1824), (12)(34), (1423)}. The only remaining self-conjugate sub- 
groups are G, and G. 

3. If a group contains all the circular substitutions on m+2 letters, it 
contains all the circular substitutions on m letters. Hint: 
(123...mm4+1m-+2)(mm-1...382m4+21m+1)=(123...m—1 Mm). 

4, Compute directly the function ¢,° of § 34 as follows: 

Py =13 +2 y3 +24) + Ox yw Ly + 80(xy7X, +1 4X4" +.%yX) + 3w?(Xyxy? +0y7x; +13”) 
= 2,3 +223 + 25° + Ox ,Lyes — 3 (Nyy + LyLy? +0473 + WyLy” + L273 + Lys” —3Vv —34, 
since 

1? —LyLq? +4424” — 11+ 12°, — TL” = — (1-22) (#2 —2s) (%—2,) =—4, 
Twice the remaining part of ¢,3 equals 2c,3—9c,c,+27c, by § 3. 

5. Compute directly the coefficients in § 36 as follows: 

t?+t.?+t,? =3 2 2;? —2 5x7; =3a?—8b, 
Lily = DU 2) Ly lok, — 22 ( Xo +-2,° + 2,7) 
=227,34+222,0,%,— 24; 2;? =4ab —8c—a? 











* For the simpler demonstration by Wantzel, see Serret, Algébre, II, 4th 
or 5th Edition, p. 512, 


SECOND PART. 


GALOIS’ THEORY OF ALGEBRAIC EQUATIONS. 


CHAPTER VY. 
ALGEBRAIC INTRODUCTION TO GALOIS’ THEORY. 


49. Differences between Lagrange’s and Galois’ Theories. Here- 
tofore we have been considering with Lagrange the general equation 
of degree n, that is, an equation with independent variables as 
coefficients and hence (see page 101) with independent quantities 
L,; Uye++, ty aS roots. Hence we have called two rational 
functions of the roots equal only when they are identical for all 
sets of values of 2,,..., %n. 

But for an equation whose roots are definite constants, we 
must consider two rational functions of the roots to be equal when 
their numerical values are equal, and it may happen that two 
functions of different form have the same numerical value. 

Thus the roots of x3+27?+2+1=0 are 


“,=—1, ~=+1, %=-1 (=Vearp 


Hence the functions 2,’, x,’, and x, are numerically equal although 
of different form. We may not apply to the equation 2x,?=2,? 
the substitution (7,77), since 2,?2,”. Again, the totality of the 
substitutions on the roots which leave the function z,? numerically 
unaltered do not form a group, since the substitutions are J, (2,23), 
(X05), (4X22). 

42 


Suc. 50] THEORY OF ALGEBRAIC EQUATIONS. 43 
Again, the reots of r*+1=0 are 


w= 6, =, %——é, = —te (=). 

/2, 
Hence v,?=e?=1, 2,2,=¢«7=1. The functions xz,’ and 2,2, differ 
in form, but are equal numerically. Also, 2,? equals 2,4”, but differs 
from 2,” and x,°._ The 12 substitutions which leave 2,? numerically 
unaltered are ,(23),(24),(34),(234) ,(243),(13),(13)(24),(213),(413), 
(4213), (4182), the first six leaving x,’ formally unaltered and the 
last six replacing z,? by x,’.. They do not form a group, since the 
product (13)(23) is not one of the set. 

There are consequently essential difficulties in passing from 
the theory of the general equation to that of special equations. 
This important step was made by Galois.* 

In rebuilding our theory, special attention must be given to 
the nature of the coefficients of the equation under discussion, 


(1) a” —¢,2"-14¢,4"—-*7— ... +(—1)"c,=0. 


Here ¢,,...,¢€n may be definite constants, or independent 
variables, or rational functions of other variables. Whereas, in 
the Lagrange theory, roots of unity and other constants were 
employed without special notice being taken, in the Galois theory, 
particular attention is paid to the nature of all new constants 
introduced. 

50. Domain of Rationality. To specify accurately what 
shall be understood to be a solution to a given problem, we must. - 
state the nature of the quantities to be allowed to appear in the 
solution. For example, we may demand as a solution a real num- 


* fivariste Galois was killed in a duel in 1832 at the age of 21. His chief 
memoir was rejected by the French Academy as lacking rigorous proofs. 
The night before the duel, he sent to his friend Auguste Chevalier an account 
of his work including numerous important theorems without proof. The 
sixty pages constituting the collected works of Galois appeared, fifteen years 
after they were written, in the Journal de mathématiques (1846), and in 
(uvres mathématiques D’ EVARISTE GALOIS, avec une introduction par 
M. Emile Picard, Paris 1897. 


Ad ALGEBRAIC INTRODUCTION. [Cu. V 


ber or we may demand a positive number; for constructions by 
elementary geometry, we may admit square roots, but not higher 
roots of arbitrary positive numbers. In the study of a given 
equation, we naturally admit into the investigation all the irra- 
tionalities appearing in its coefficients; for example, V3 in con- 
sidering 2?+(2—5V3)e+2=0. We may agree beforehand to 
admit other irrationalities than those appearing in the coefficients. 

In a given problem, we are concerned with certain constants 
or variables 


(2) R’, R”,..., Rw 


together with all quantities derived from them by a finite number 
of additions, subtractions, multiplications, and divisions (the 
divisor not being zero). The resulting system of quantities is 
called the domain of rationality * (R’, R”’,..., R™). | 


Exampte 1. The totality of rational numbers forms a domain. It is con- 
tained in every domain Ff. For if w be any element ~0 of R, then w+w=1 
belongs to R; but from 1 may be derived all integers by addition and sub- 
traction, and from these all fractions by division. 

Exampte 2. The numbers a+bi, where i= —1, while a and b take 
all rational values, form a domain (7). But the numbers a+bi, where a and 
b take only integral values do not form a domain. 


DEFINITION. An equation whose coefficients are expressible 
as rational functions with integral coefficients of the quantities 
R’, R”’,..., R will be said to be algebraically solvable (or solvable 
by radicals) with respect to their domain, if its roots can be de- 
rived from R’, R”,... by addition, subtraction, multiplication, 
division, and extraction of at root of any index, the operations 
being applied a finite number of times. 

51. The term rational function is used in Galois’ theory only 


* Rationalitdtsbereich (Kronecker), Korper (Weber), Field (Moore). 

+ If we admitted the extraction of all the pth roots, we would admit the 
knowledge of all the pth roots of unity. This need not be admitted in Galois’ 
theory (see § 89, Corollary). 


Src. 52] THEORY OF ALGEBRAIC EQUATIONS. 45 


in connection with a domain of rationality R. An integral rational 


function for F of certain quantities wu, v, w, ...1S an expression 
(3) Die Gap EU ai, 
, i, i k, ae, 
\ 
where 7,7, k,... are positive integers, and each coefficient Cy,... 


is a quantity belonging to R. The quotient of two such functions 
(3) is a rational function for R. 
Thus, 3u+ 2 is a rational function of u in (2), but not in (1). 
52. Equality. As remarked in § 49, two expressions involving 
only constants are regarded as equal when their numerical values 
are the same. Consider two rational functions 


plu, v; UW, ys oe Pu, v; W, 8 3 


with coefficients in a domain R=(R’, R’”,..., R®). In case R’, 
R’’,... are all constants, we say that ¢ and ¢ are equal if, for 
every set of numerical values u,, v,, w,,... which u, v, w,... can 
assume, the resulting numerical values of ¢ and ¢ are equal. In 
case R’, R”’,..., R” depend upon certain independent variables 
r,r’,..., 7™, we say that ¢ and ¢ are equal if, for every set of 
numerical values which u,v, w,..., 7, 7”,..., 7™ may assume, 
the resulting numerical values of ¢ and ¢ are equal. When not 
equal in this sense, ¢ and ¢ are said to be distinct or different. 

For example, if w and v are the roots of x?+2or+1=0, the functions 
u+v and —2puv are rational functions in the domain (p), and these rational 
functions are equal. 


DerFIniTion. A rational function ¢(2%,,..., %) is said to be 
unaltered by a substitution s on %,..., x, if the function 
Pp(%,---, Tn) is equal to @ in the sense just explained. For 


brevity, we shall often say that ¢ then remains numerically un- 
altered by s. If 2,, 2,..., 2 are independent variables, as in 
Lagrange’s theory, and if ¢, is identically equal to ¢, 1.., for all 
values of 2,,..., 2m, we say that ¢ remains formally unaltered 
by s. For examples, see § 49. 

58. The preceding definitions are generalizations of those 
employed in the Lagrange theory. The so-called general equation 


46 ALGEBRAIC INTRODUCTION. . [Cu. V 


of degree n may be viewed as an extreme case of the equations (1) 
whose coefficients ¢,,..., C, are rational functions in the domain 
(R’, R’,..., R™). In fact, since its coefficients are independent 
variables belonging to the domain, they may be taken to replace 
an equal number of the quantities R’, R’”,... defining the do- 
main, so that the general equation appears in the form 


an Reeth Re 2+ 2. + RO, 


Its roots are likewise independent variables (p. 101), so that two 
rational functions of the roots are equal only when identically equal. 

54. Reducibility and irreducibility. An integral rational func- 
tion F(x) whose coefficients belong to a domain R is said to be 
reducible in & if it can be decomposed into integral rational factors 
of lower degree whose coefficients likewise belong to #; irreducible 
in F if no such decomposition is possible.* 

ExampteE 1. The function 2?+1 is reducible in the domain (7) since it 


has the factors x+7 and x+17, rational in (7). But 2?+1, which is a rational 
function of x in the domain of rational numbers, is irreducible in that domain. 


cee ir x*+1 is reducible in any domain to which either nv 4 2, or V 7 


or 2, Or cae belongs, but is irreducible in all other domains. In fact, its 
linear factors are c+e, r+ie=x+e?; while every quadratic factor is of the 
form x?+1, or z*+ar+1, a =+2. 


If (x) is reducible in R, F(x)=0 is said to be a reducible 
equation in #; if f(x) is irreducible in R, F(x)=0 is said to be 
an irreducible equation in R. 

55. THroreM. Let the equations F(x)=0 and G(x)=0 have 
their coefficients in a domain R and let F(x) =0 be irreducible in R. 
If one root of F'(x)=0 satisfies G(x) =0, then every root of F(x)=0 
satisfies G(x)=0 and F(x) vs a divisor of G(x) in R. 

After dividing out the coefficients of the highest power of z, let 


ee (x—¢,)(a—&,) . ety G(x) =(@#— 1m)... (@— Hm). 








* A eEod to decompose a given integral function by a finite number 
of rational operations has been given by Kronecker, Werke, vol. 2, p. 256. 


Src. 55] THEORY OF ALGEBRAIC EQUATIONS. 47 


At least one ¢ equals an 7. Let €,=7,,..., &.=7,, while the 
remaining ¢’s differ from each 7. Then the function 


B(ry=(e—f,)... (x—€,) =(2—7,) .. . (42 — 7) 


is the highest common factor of F(x) and G(«#). But Euclid’s 
process for finding this highest common factor involves only 
the operation division, so that the coefficients of A(x) are 
rational functions of those of F(a) and G(x) and consequently 
belong to the domain R. Hence F(x%)=A(x)-Q(x), where A(x) 
and Q(x) are integral functions with coefficients in R. Since F(x) 
is irreducible in FR, Q(x) must be a constant, evidently 1. Hence 
F(«)=H(z), so that F(x) is a divisor of G(x) in R. 

Corotuary I. If G(x) is of degree en—t, then G(xz)=0. A 
root of an irreducible equation in # does not satisfy an equation 
of lower degree in Rf. 

CorouuarRy II. If also G(x)=0 is irreducible, then G(x) is a 
divisor of I(x), as well as I’(x) a divisor of G(x). If two irreducible 
equations in FR have one root in common, they are rdentical. 


CHAPTER VI. 
THE GROUP OF AN EQUATION, 


EXISTENCE OF AN 7!-VALUED FUNCTION; GALOIS’ RESOLVENT. 


56. Let there be given a domain # and an equation 


(1) f(x) =a" —c,0"—1+6,2"-7— ... +(—1)"en=0, 
whose coefficients belong to R. We assume that its roots 4%, 
Yo,..+, %m are all distinct.* It is then possible to construct a 


rational function V, of the roots with coefficients in R such that 
V, takes n! distinct values under the m! substitutions on 2, ,..., Xn. 
Such a function is 


V,=M,2,+MH+ ... +MnIn, 


if m,,...,M%, are properly chosen in the domain R. Indeed, 
the two values V, and Vj», derived from V, by two distinct sub- 
stitutions a and 0b respectively, are not equal for all values of 
M.,.++,Mn, Since 2,,...,%, are all distinct. Siygise eee 
possible to choose values of m,,...,m™, in R which satisfy none 
of the 4n!(n!—1) relations of the form V,=Vj,,. 

Then from an equation Vy,-=V, will follow a’ =a. 


As an example, consider the equation x*+2?+a+1=0, with the roots 
x=—1, t= +1=+/—I, X3= —1, 
and let R be the domain of all rational numbers. The six functions 
‘ 4 
—™M,+1M,—1M3, —M,—1M,+1Mz, 1M,—M,—MM,, , 
—1Mm,+1m,—M,,° —im,—mM,+1mM;, 1m,—iIm,—M., 


* Equal roots of F(x) =0 satisfy also F'’(x) =0, whose coefficients likewise 
belong to R, and consequently also H(x) =0, where H(z) is the highest com- 
mon factor of F(«) and F’(x). If F(x) +H (x) =Q(a), the equation Q(x) =0 
has its coefficients in R and has distinct roots. After solving Q(x) =0, the © 
roots of F(x) =0 are all known. 


48 


Src. 57] THEORY OF ALGEBRAIC EQUATIONS. 49 


arising from the 3! permutations of x,, x,, x3, will all be distinct if no one of 
the following relations holds: 
M,—m,=0, m—m,=0, m,—m,=0, 
(i+1)m,—2im,+(C—1)m, =0, (@—1)m,+(0¢+1)m,—2im, =0, . 
(t—1)m,—21m,+(¢+1)m, =0, (1+1)m,+(—1)m,—2im, =0, 
—2im,+ (t—1)m,+ (1+ 1)m, =0, —2im, +(¢7+1)m,+ (i—1)m,=0, 
of which the last six differ only by permutations of m,, m,, m;. We may, 
for example, take m,;=0 and any rational values ~0 for m, and m, such 
that m,~cm,, where c is 1, +7, 1 +7,4(1+7). Thus mz,4+2, is a six-valued 
function in FR if m, is any rational number different from 0 and 1. 

[In the domain (7), we may take m,x,+2,, where m,~0, 1, +2, 141, 


2(147).] 
57. The n! values of the function V, are the roots of an equation 
(4) PN IeatV SV iV —¥)... Va Va, 


whose coefficients are integral rational functions of m,,..., Mn, 
C;, +++, €n With integral coefficients and hence belong to the domain 
R (§ 50). If F(V) is reducible in R, let F,(V) be that irreducible 
factor for which /’,(V,)=0; if F(V) is irreducible in R, let F,(V) 
be F(YV) itself. Then 
(5) FY(V)=0 
is an wrreducible equation called the Galois resolvent of equation (1). 
Recurring to the example of the preceding section, take 
Ve=7,—4)," Vo=—2,—13, - Ve=%3— 7}. 
Then the six values of V, are +V,, +V., +V;, where 
Watt l, V¥,=21, Vyao~itl. 
The equaticn (4) now becomes 
eet CHV ?—V,7)(V?2—V 5°) =(V?—2)(V? +4)(V? +22) 
=V°+4V4+4V?+16=0. 
The irreducible factors of F(V) in the domain of rational numbers are 
Weta =(V—V,\(V+V,), V2-2V+2=(V—Vi(V -F,;), 
V242V4+2=(V+V,(V+V,). 
The Galois resolvent (5) is therefore 
FAV) =V2—2V +2=0. 
[For the domain (7), the Galois resolvent is V-—V,=V—i—1=0.] 


50 THE GROUP OF AN EQUATION. [Cu. VI 


58. THrorem. Any rational function, with coefficients im a 
domain R, of the roots of the given equation (1) ts a rational function, 
with coefficients in R, of an n!-valued function V,: 


(6) | P(X, Tq). +» Ln) = O(V,). 
Let first the coefficients c,,...,¢, In equation (1) be arbitrary 
quantities so that the roots 2,,...,2, are independent variables. 


We may then apply the proof in §$ 31 of Lagrange’s Theorem, 
taking for ¢ the function V, which is unaltered by the identical 
substitution alone, and obtain a relation 


(6”) i A el: 
where F’(V) is the derivative of /’(V) defined by (4). We next 
give to c,,..., Cy, their special values in #, so that z,, . . ., %, become 


the roots of the given equation. Since F’(V,)40, relation (6’) 
becomes the desired relation (6), expressing ¢ as a rational function 
of V, with coefficients in R. 

CoROLLARY. I} s be any substitution on the letters x,,..., Xn, then 


(7) Ps(Xy, Lo, .--, tn) =O(V 5), 
provided no reduction* in the form of ®(V,) has béen made by 
means of the equation I'(V,)=0 of § 57. 


As an example, we recur to the equation 23+2?+2+1=0, and seek an 
expression for the function ¢=2, in terms of V,=2,—a,. Then 


F(V) =V°+4V!44V?+16, F’(V)=6V5+16V?+8V, 





) 7 =K v5 § a) vy - ks v3 v3 vy 
VY—FW) 1 Voy, V+V, 1 V—V, VV, ee 

= —2V°—4V4—12V3—8V?—16V —48, 
upon setting 7,= —1, x,=1, 7,=—1, V;=71+1, V,=21, V;=—7i+1. Hence 
_ AV) _ 2 5-4 12) See 
eFC 5 6V,5+16V,°+8V, 
In verification, we find that 


AV ,) =AQti+1) = —481—-16,  F’(V,) =161—48, O(V,) =1=2,. 


Ly 





= 0(V;,). 





* That such a reduction invalidates the result is illustrated in the example | 


of § 59. 


Src. 59] THEORY OF ALGEBRAIC EQUATIONS. 51 


In view of the corollary, we should have 
%,=0(—V,), 1=9(V,), 2,=9(V;), 2%,=9(—V,), 2,=9(—YV,). 
To verify these results, we note that 


16.—48 _ N80! _ =80 
ro ge a Bor cae 
while O(V;) and O(V,), 9(—V;), and 9(—V;,), xz, and 2, are conjugate 
imaginaries, and 2, is real. 


o-V)= 


59. As a special case of the preceding theorem, the roots of the 
given equation are rational functions of V, with coefficients in R: 


(8) 1,=0,(V,), t2=¢2(V1),..-, In=Gn(Vj). 


Hence the determination of V, is equivalent to the solution of the 
given equation. 

Since each V, is a rational function of 2,, .. ., , with coefficients 
in R, it follows that all the roots of the Galois resolvent are rational 
functions with coefficients in R of any one root V,. 


Exampte. For the equation x*+2*+x+1=0, and V,=2,—2, we have 


2,=—1; 2,=V,—1, %3=—V,+1, V,=2V,-2, V,=—V,+2. 


Although x, and V,—1 arenumerically equal, the functions z,and —V,—1, 
obtained by applying the substitution (z,2,), are not equal. The relation 
2,=V,—1 is a reduced form of z,=@(V,), obtained in virtue of the identity 
V ?—2V,+2=0 ($57). Thus 


—2V,5—4V,*—12V,’—8V,?—16V, —48 a) —48V, +32, 
6V P+ 167° +8V, a 16V, — 64, 


—48V,+32_(—3V,+2)(Vi+2)_—3V—4V,+4__—10V,+10 
16V,—64 a (V,—4)(V,+2) V’—2V,-8 —10 





=V,-1. 


It happens, however, that the equality 7,=V,—1 leads to an equality 
x,=V,;—1=—V,+1 upon applying the substitution (a,7;). The fact that 
the identical substitution and (2,7;), but no other substitutions on 2, 2, 23, 
lead to an equality when applied to z,=V,—1 finds its explanation in the 
general theorems next established. 


52 THE GROUP OF AN EQUATION. (Cu. VI 


THE GROUP OF AN EQUATION. 
60. Let the roots of Galois’ resolvent (5) be designated 


(9) ve Va; a7 Foe ey Var 
the substitutions by which they are derived from V, being 
(10) |e AY ey get h 


These substitutions form a group G, called the group of the given 
equation (1) with respect to the domain of rationality R. 

The proof consists in showing that, if r and s are any two of 
the substitutions (10), the product rs occurs among those substi- 
tutions. Let therefore V, and V, be roots of (5). Then 


eV le 0: 
Now VJ, is a rational function of V, with coefficients in R: 
(11) V,=YV,), 


the function 0 being left in its unreduced form as determined in § 58. 
Hence F'{0(V,)]=0, so that one root V, of the equation (5) irre- 
ducible in R satisfies the equation 


(12) {AV )]=9, 
with coefficients in R. Hence (§ 55) the root V, of (5) satisfies (12). 
io. OE) | =0- 
In view of the corollary of § 58, it follows from (11) that 
(V,)s=Vrs=A(Vs). 
Hence [’,(V;s)=0, so that V,, occurs among the roots (9). 


ExamptLeE, For the equation v*+2?+2-+1=0 and the domain R of rational 
numbers, the Galois resolvent was shown in § 57 to be V?—2V +2=0, having 
the roots V; and V;._ Since V; was derived from V, by the substitution (2,2), 
the group of the equation x*+2?+2+1=0 with respect to R is {J, (apa,)}. 

For the domain (7), the Galois resolvent was shown to be V—V,=0. 
Hence the group of the equation with respect to (7) is the identity. 


Src. 61] THEORY OF ALGEBRAIC EQUATIONS. 53 


61. The group G of order N of the equation (1) with the roots 
2%, X,.. +, Xn possesses the following two fundamental properties: 

A. Every rational junction (2, %,..., tm) of the roots which 
remains unaltered by all the substitutions of G lies in the domain R. 

B. Every rational function $(4,, %,...,%n) of the roots which 
equals a quantity in R remains unaltered by all the substitutions of G. 

By a rational function 6=¢(a,,..., 2») of the roots is meant 
a rational function with coefficients in R. Then by § 58 


(18)  P=O(V,), Pa=P(Va), Po=P(Vs),.--, P= O(V2), 


where @ is a rational function with coefficients in R. 
Prooj of A. If d6=¢a=dp= .. . = Gj, it follows from (18) that 


b= TOV) FOV) OM) +... +00}. 


The second member is a symmetric function of the N roots (9) of 
Galois’ resolvent (5) and hence is a rational function of its coeffi- 
cients which belong to R. Hence ¢ lies in R. 

Proof of B. If ¢ equals a quantity r lying in R, we have, in 
view of (138), the equality 


0(V,)—r=0. 
Hence J, is a root of the equation, with coefficients in &, 
(14) 0(V)—r=0. 


Since one root V, of the irreducible Galois resolvent equation (5) 
satisfies (14), all the roots V,, Va,..., Vz of (5) satisfy (14), 
in view of § 55. Hence 


fe j—-7—0, O(V,)—r=0, ..., OVp)—r=0. 


It therefore follows from (13) that 6=¢d.=q¢)=...=¢i. Hence 
¢ remains unaltered by all the substitutions of G. 

62. By arational relation between the roots 2, ..., % 1s meant 
an equality f(2,,...,%n)=¢(%,,..-,%n) between two rational 
functions, with coefficients in R. Then d6—¢is a rational function, 


54 THE GROUP OF AN EQUATION. [Cu. VI 


equal to the quantity zero belonging to R, and therefore (by B) 
is unaltered by every substitution s of G. Hence d,—¢,=$—¢=0, 
so that d6,=¢,. Hence the result: 
Any rational relation between the roots remains true if both 
members be operated upon by any substitution of the group G. 
ExampLe. For the domain of rational numbers, it was shown in § 60 


that the equation z°+2?+2+1=0 has the group {J, (a,,)}. The rational 
relation (§ 59, Example) 


leads to a true relation 2,=x,—x,—1=V,—1 under the substitution (425). 
If we apply (2,2,), we obtain a false relation xz, =2,—z,—1. 


63. THrorEM. Properties A and B completely define the group G 


of the equation: any group having these properties vs rdentical with G. 
Suppose first that we know of a group 


G’={I, a’, ,..., m} 


that every rational function of the roots 2,,..., 2%, which remains 
unaltered by all the substitutions of G’, liesin R. The equation 


F(V)=(V—V)(V—Va)(V—Vy)..- (V—-Va)=0 


has its coefficients in R since they are symmetric functions of 
V,, Va’,.--, Vm and therefore unaltered by the substitutions 
of G’. Since /’’(V)=0 admits the root V, of the irreducible Galois 
resolvent (5), it admits all the roots V,, Va,..., V, of (5). Hence 
I, a,..., l occur among the substitutions of G’, so that @ is a 
subgroup of G’. | 

Suppose next that we know of a group 


GAT We UG ag 


that every rational function of 2,,..., X, which lies in R-remains 
unaltered by all the substitutions of G’’. Then the rational function 
F(V,), being equal to the quantity zero lying in R, remains un- 
siteredsby-a’ , 0571s... wie. a0 Lua 


0=F (VV) =F (Va)=P (Vor = ... =F Vir) e 


Src. 64] THEORY OF ALGEBRAIC EQUATIONS. > 55 


Hence V,;Va”,..., V-’ occur among the roots V,, Va,..., V; of 
F(V)=0. Hence G” is a subgroup of G. 
If both properties hold for a group, G’=G”; then G’ contains 
G as a subgroup and G’ is a subgroup of G. Hence G’=G”=G. 
It follows that the group of a given equation for a given domain 
as unique. In particular, the group of an equation is independent 
of the special n!-valued function V, chosen. 


EXAMPLE. For the equation x?+x?+x+1=0 and the domain F of all 
rational numbers, the functions +V,, +V., +V; of y 57 are each 6-valued 
Employing V,, we obtain the Galois resolvent 


(V—V,)(V —V;) =V?—2V +2=0 
and the group {/, 2,2,)}. Evidently no change results from the employment 
of V;. If we employ either —V, or —V;, we obtain the Galois resolvent 
(V+Vi)(V+V3) =V?+2V+2=0 
and the group {J, (x,7,)}. If we employ either V, or —V., we get 
(V—V.)(V+V,) =V?+4=0. 
Since V,=2z,—2;, the substitution replacing V, by —V, is (a3), so that 
the group is again {J, (2,.,)}. 


ACTUAL DETERMINATION OF THE GROUP G OF A GIVEN EQUATION. 


64. Group of the general equation of degree n. Its coefficients 
C1). ++, are independent variables, and likewise its roots (p. 101). 
We proceed to show that, jor a domain R containing the coefficients 
and any assigned constants, the group of the general equation of 
degree n 1s the symmetric group Gp. It is only necessary to show 
that the Galois resolvent /’,(V)=0 is of degree n!. In the relation 
F(V,)=0, we replace V, and the coefficients ¢,..... ¢, by their 
expressions in terms of x,,...,2,. Since the latter are independent, 
the resulting relation must be an identity (see p. 101) and hence - 
remain true after any permutation of 2,,...,%. By suitable 
permutations, V, 1s changed into V,,.... V»; in turn, while ¢,,..., ¢n, 
being symmetric functions, remain unaltered. Hence /,(V,)=0, 

., F(Vn1)=0. Hence F,(V)=0 has n! distinct roots. 

Another proof follows from § 63 by noting that properties A 
and B hold for the symmetric group G,,; when 2,,..., %» are inde- 


56 THE GROUP OF AN EQUATION. [Cu. VI 


pendent variables. Thus A states that every symmetric function 
of the roots is rationally expressible in terms of the coefficients. 

65. To determine the group of a special equation, we usually 
resort to some device. It is generally impracticable to construct 
an n!-valued function and then determine the Galois resolvent (5) ; 
or to apply properties A and B directly, since they relate to an 
infinite number of rational functions of the roots. Practical 
use may, however, be made of the following lemma, involving a 
knowledge of a single rational function: 

Lemma. J} a rational function (a,,...,%pn) remains formally 
unaltered by the substitutions of a group G’ and by no other substi- - 
tutions, and if ¢ equals a quantity lying in the domain R, and 7 the 
conjugates of & under Gy, are all distinct, then the group of the given 
equation for the domain R is a subgroup of G’. 

In view of the first part of § 68, it is only necessary to show 
that every rational function $(2,,..., 2%), which remains numeri- 
cally unaltered by all the substitutions of G’, ies in R. If G’ is 
of order P, we can set 


b=btb.t ... +4P); 


so that ¢ can be given a form such that it is formally unaltered by 
all the substitutions of G’. Then, by Lagrange’s Theorem (§ 31), 
¢@ is a rational function of ¢ and hence equals a quantity lying 
in fF. 

EXAMPLE 1. To find the group of z?—1=0 for the domain R of all rational 
numbers. The roots are 

a=1, %=3(—1++/-8), ,=4(—1—*/—3). 

Taking ¢ =2,, it follows from the lemma that Gis a subgroup of G’ = {J, (a2,)}. 
Since x, does not lie in R, Gis not the identity (property A). Hence G=@’. 

EXAMPLE 2. To find the group G of y3—7y+7=0 for the domain R of 
all rational numbers. 

For the cubic y?+py+q=0, we have (§ 2) 

D=(Y1—Y2)"(Y2—Ys) "(Ys Yi)? = — 279? —4p* 
For p= —7, g=7, we get D=7*. Hence the function 
$= (Y1—Y2) Y2—Ya) Ys —Yy) 


Src. 65 THEORY OF ALGEBRAIC EQUATIONS. 57 


has a value +7 lying in #& and its conjugates ¢ and —¢ under G;, are distinct. 
By the lemma, G is therefore a subgroup of the alternating group G;, and 
hence either G, itself or the identity G,. Now, if the group of the equation 
were G,, its roots would lie in R. But * a rational root of an equation of 
the form y?—7y +7 =0, having integral coefficients and unity as the coefficient 
of the highest power, is necessarily an integer. By trial, +1, +7 are not 
roots. Hence the roots areall irrational. Hence the group G is G. 

EXAMPLE 3. Find the group of z'+1=0 for the domain of rational 
numbers. 

We seek a rational function of the roots x, 7, 73, x, which equals a rational 


number. Let us try the function y,=2%,2,+2%,2, Specializing the result — 
holding for the general quartic equation (§ 4), we find that, for the quartic 
x*+1=0, the resolvent equation (16) for y, is 


y>—4y=0. 
By a suitable choice of notation to distinguish the roots x:, we may set 
Yi=—2, Y2=0, Ys=+4+2. 
Hence y, equals a rational number and its conjugates under G,, are all distinct. 
Hence G is a subgroup of G;, the group to which 2,2,+2,7, belongs formally 
(§ 21). Similarly, by considering the conjugate functions y,=2,7,+2,%,, 


and y;=2,%,+2,2,, we find that G is a subgroup of Gj and Gj’. Hence G 
is a subgroup of G,(§ 21). Hence G is G,, G,, 

— Gy =[L, (& 4X2) (X32) }, Gh [T, (2123) (2%) }, or GE = [T, (ayy) (as) }. 

Now G#G,, since no root of «!+1=0 is rational. 

If G,, consider ¢,=2,+%,—2,—2, For the general quartic equation 
x*+ax>+bx?+cx+d=0, we have t,?=a’—4b+4y, by §5. Hence, for 
z'+1=0, t,?=—8. Since ¢, is not rational, G#G. 

If G;’, consider t,=2,+2,—2,—2;. In general, t,2=a?—4b+4y,;. Here 
t,2,=+8. Since é, is not rational, G#G,’. 

» If G), consider ,=2,+2,;—2,—2, In general, é,2>=a?—4b+4y,. Here 
t,2=0. Since a conjugate —é, of t, equals ¢,, no conclusion may be drawn 
from the use of ¢,. But ¢=2,7,;—2,2, is unaltered by Gj. Now 


df? = (2103+ 2,0,)’ —427,0,0,0,=Y,” -—4 = —4, 
Hence ¢ is not rational, so that GG}. 
The group of z*+1=0 for the domain of rational numbers is therefore G. 


EXERCISES. 
Find for the domain of rational numbers the group of 
1, z?+2?+2+1=0 (using the lemma, § 65). 
2. (x—1)(4+1)(x—2) =0. 


* Dickson, College Algebra (John Wiley & Sons), p. 198. 





58 THE GROUP OF AN EQUATION. [Cu. VI 


3. 23—2=0. [2,, X, v7, and (1, —2,)(x,—23)(%3—2,) are irrational. ] 

4, e4+a3+2?+2+1=0 with roots 7,=¢, %,=¢«7, 7%,=e*, 7,=6*, where 
¢ is an imaginary fifth root of unity. Since the resolvent for 2,27,+ a2, is 
y®—y?—3y+2=0 with the roots 2, $(—1+/5), G is a subgroup of G%. 
The latter has the subgroup C,= {J, (1234), (13)(24), (1432) }, to which belongs 
fy =2 12%, 42,20, +2,°u,+2,72, Here d,=e'+ e8+¢e+e?=—1 is rational. The 
six conjugates to ¢, under G,, are distinct; they are obtained from ¢, by 
applying I, (12)(34), (12), (14), (23), (34); their values are —1, 4, 1+2¢e+?, 
14+2e8+e4 142e?+c¢, 1424+? respectively. Hence G is a subgroup 
of C,. To G,={J, (13)(24)} belongs 


(x,—2x, +22, —ix,)? =(1+2i)(e? + 68 — et —e) = 4.4/5 (1421). 


Hence G#¥G}. Evidently G#G,. Hence G=C,. 

5. Show that, for the domain (1, 7), the group of #!+1=0 is G4. 

6. Show that, for the domain (1, w), w=imaginary cube root of unity, 
the group of z?’—2=0 is C,;={I, (x,12%3), (2427) }. 

Hint: (2, +w2,+wx,)? and (7,+w?x,+w2,)? have distinct rational values. 


TRANSITIVITY OF GROUP; IRREDUCIBILITY OF EQUATION. 


66. A group of substitutions on n letters is transitive if it 
contains a substitution which replaces an arbitrarily given letter 
by another arbitrarily given letter; otherwise the group is intran- 
sitive. 

Thus the group G,= {I, (272) (a3%,), (24%) (Xo%%,), (2%,)(%_%3)} is transitive ; 
I replaces x, by 2,, (x,%_)(x,%,) replaces x, by 22, (1,%3)(%,%,) replaces x, by Zz, 
(x,x,)(x,%,) replaces x, by x, Having a substitution s which replaces 2, 
by any given letter z; and a substitution ¢ which replaces x, by any given 
letter 2;, the group necessarily contains a substitution which replaces 2; 
by 2;, namely, the product s—1. 

The group H,={J, (x,%), (x3%), (,%,)(%,%,)} is intransitive. 


67. THEOREM. The order of a transitive group on n letters is 
divisible by n. 

Of the substitutions of the given group G, those leaving 2, 
unaltered form a subgroup H={I, h,,..., h,}. Consider a rect- 
angular array (§ 28) of the substitutions of G with those of H in 
the first row, choosing as g, any substitution replacing x, by a, 
as g, any substitution replacing x, by x,, etc. Then all the sub- 
stitutions of the second row and no others will replace a, by 2,, 


> 


a ok aso 


= ne en a 


gw gr pe coe ee 


af 


———— ain LR 


Src. 68] THEORY OF ALGEBRAIC EQUATIONS. 89 


all of the third row and no others will replace xz, by x, ete. Since 
G is transitive, there are v=n rows. But the order of G is 
divisible by v (§ 26). 


Examples of transitive groups: G;@), G,@), G9, G4, G@, GO. 


The least order of a transitive group on 7 letters is therefore n. 
A transitive group on 7 letters of order 7 is called a regular Broup. 
Thus G,\® and G,( are regular. 

68. THrorEM. I] an equation is irreducible for the domain R, 
its group for R ws transitive; if reducible, the group is intransitive. 

First, if f(a) =0 is irreducible in R, its group for £# is transitive. 
For, if intransitive, G contains substitutions replacing 2, by 4%, 
T>,.+-, tm, but not by rm4,,..-, Ln, the notation for the roots 
being properly chosen. Hence every substitution of G permutes 
Z14,-++, Um amongst themselves and therefore leaves unaltered 
any symmetric function of them. Hence the function g(x)= 
(w—2,)(t—2X)...(*—X») has its coefficients in R, so that g(x) 
isa rational factor of f(x), contrary to the irreducibility of f(z). 

Let next /(x) be reducible in & and let g(x) =(a—2,) .. . (x—Xm) 
be a rational factor of j/(x), m being<n. The rational relation 
g(x,)=0 remains true if operated upon by any substitution of G 
(§ 62). Hence no substitution of G can replace x, by one of the 
rootS mij,---,%n; for, if so, g(x)=0 would, have as root one of 
the quantities %4+4,,...,2n, contrary to assumption. Hence G is 
intransitive. 


EXAMPLE 1. The equation x?—1=0 is reducible in the domain R of 
rational numbers; its group for R is {J, (x,73)} by § 65, Ex. 1, and is intran- 
sitive. <A like result holds for x?+2?+x2+1=0 (§ 60). 

EXAMPLE 2. The equation y*—7y+7=0 is irreducible in the domain 
R of rational numbers, since its left member has no linear factor in R (§ 65, 
Ex. 2). Hence its group for R is transitive. By § 65, the group is G,(). 

EXAMPLE 3. The equation z*+1=0 is irreducible in the domain F& of 
rational numbers (§ 54, Ex. 2). Hence its group for # is transitive, and 
so is of order at least 4. We may therefore greatly simplify the work in 
§ 65, Ex. 3, for the determination of the group G. 

Examp_Le 4, The equation x*+1=0 is reducible in the domain (1, 2). 
Its group Gj is intransitive (see Ex. 5, page 58). 


60 THE GROUP OF AN EQUATION. [Cu. VI 


RATIONAL FUNCTIONS BELONGING TO A GROUP. 


69. THrorEM. Those substitutions of the group G of an equation 
which leave unaltered a rational function & of its roots form a group. 

Let I, a, b,..., k be all the substitutions of G which leave ¢ 
unaltered (in the numerical sense, § 52). Apply to the rational 
relation 6=¢, the substitution b of the group G. Then (§ 62) 
dv=da. Hence da=¢, so that the product ab is one of the 
substitutions leaving ¢@ unaltered. Hence the substitutions J, 
a,b,...,k form a group H. 

No matter what group ¢ belongs to formally (§ 21), we shall 
henceforth say that ¢ belongs to the group H, a subgroup of G. 


Examp.Le. For the domain RF of rational numbers the group of #4+1=0 is 
Gi, = [L, (© 1X2) (@ 54), (1423) (L2X,4), (1X4) (XpXq) }, 
by § 65, Ex. 3. Of the 12 substitutions which leave x,? numerically unaltered 
(§ 49), only IJ and (2,2%;)(x,%,) occur in G,. Hence the function 2,? of the 
roots of «*+1=0 belongs to the group {J, (x7) (aa) }. 


70. THroreM. J} H is any subgroup of the group G of a given 
equation for a domain hk, there exists a rational function of tts roots 
with coefficients in R which belongs to H. 

Let V, be any n!-valued function of the roots with coefficients 
in R (§ 56). Let V,, Va,..., Vz be the functions derived from 
V, by applying the substitutions of H. Then the product 


—P=(0—Vi)(0— Va) - - - (o— Va) 


in which ¢ is a suitably chosen quantity in R, is a rational function 
of the roots with coefficients in & which belongs to H (compare 
$25): 

71. THrorem. If a rational function ¢ of the roots of an equation 
belongs to a subgroup H of index v under the group G of the equation 
for a domain R, then ¢ takes v distinct values when operated upon 
by all the substitutions of G; they are the roots of a resolvent equation 
with coefficients in k, 


(15) gy=(y—- $i) (y— $2) --- Y- Pr) =0. 


Src. 72] THEORY OF ALGEBRAIC EQUATIONS. 61 


The proof that there are exactly v distinct values of ¢ under the 
substitutions of G is the same as in § 29, the term distinct now 
having the meaning given in § 52. 

Any substitution of the growp G merely permutes the functions 
d1, Yo, ..., %» (compare § 30), so that any symmetric function of 
them is unaltered by all the substitutions of G and hence equals a 
quantity in R (Theorem A, § 61). Hence the coefficients of (15) 
lie in R. 3 

Remark. The resolvent equation (15) ws trreducible in R. 

Let y(y) be a rational factor of g(y). Applying to the rational 
relation 7(¢,)=0 the substitutions of G, we get 7(¢,)=0,..., 
r(¢.)=0. Hence 7(y)=0 admits all the roots of g(y)=0, so that 


ry) =9(y). 


ExampLeE 1. For the domain fF of rational numbers, the group G of 
xe+a?+2+1=0 is {J, (x,7,)}, by § 60. The conjugates to x,—z, under G 
are J, =2,—2,, $,=X;—2X,. They are the roots of 


y? —(d,+¢2)y + Pig, =y? —2y +2 =0. 


EXAMPLE 2. For the domain (1, 7), the groupGof «+1 =0 is {I,(x,x7;)(x,7,)}, 
by Ex. 5, page 58, employing the notation of § 49 for the roots. The con- 
jugates to x, under G are ¢,=2,, ¢,=x3. They are the roots of 


y?—(e—e)yt+ e(—e) =y?—71=0. 
It is irreducible in (1, 2), since \/i=(1+0) +~/2. 


72, LAGRANGE’S THEOREM GENERALIZED BY GALoIs. If a 
rational function $(%,, X,,..., Xn) of the roots of an equation f(x) =0 
with coefficients in a domain R remains unaltered by all those sub- 
stitutions of the growp G of j(x)=0 which leave another rational 
function $(4,, L,...,Xn) unaltered, then > is a rational function of 
d with coefficients in R. 

The function ¢ belongs to a certain subgroup H of G, say of 
index v. By means of a rectangular array of the substitutions 
of G with those of H in the first row, we obtain the v distinct con- 
jugate functions ¢,,¢,,..., ¢, and a set of functions ¢,, d,..., dy, 
not necessarily distinct, but such that a substitution of G which 


62 THE GROUP OF AN EQUATION. [Cu. VI 


replaces ¢; by ¢; will replace ¢; by ¢; (compare § 31). If g(d) 
be defined by (15), then 





=a) (P+ e+ ae +e) 


is an integral function of t which remains unaltered by all the 
substitutions of G, so that its coefficients lie in & (§ 71). Taking 


¢,=¢ for t, we get P=A(Y) +9). 


For examples, see § 58. The function V, is unaltered by the identical 
substitution only, which leaves unaltered any rational function. 


REDUCTION OF THE GROUP BY ADJUNCTION. 


78. For the domain R=(1) of all rational numbers, the group of the 
equation 2°+2?+24+1=0 is G,={J, (x,73)}; while its group for the domain 
R’ =(1, 7) is the identity G, (see § 60). In the language of Galois and 
Kronecker, we derive the domain R’ =(1, 2) from the included domain R =(1) 
by adjoining the quantity 7 to the domain Rf. By this adjunction the group 
G, of «3+2?+2+1 is reduced to the subgroup G,. The adjoined quantity 7 
is here a rational function of the roots, 7=x7,=—vzs, in the notation of § 49 
for the roots. The Galois resolvent V?—2V +2=0 for R becomes reducible 
in R’, viz., (V—i—1)(V +7—1) =0. 

For the domain R=(1), the group of x*+1=0 is G,; for the domain (1, 2), 
its group is the subgroup Gj={J, (a,x3)(a,27,)}, by § 65. By the adjunc- 
tion of 7 to the domain R, the group is reduced to a subgroup Gj. Here 
(=2,? =2,? = —2x,?= —2,?=2,7,, in the notation of §49. The subgroup of 
G, to which «,? belongs is Gj. If we afterwards adjoin +/2, the roots will 
all belong to the enlarged domain (1, 7, \/2), so that the group reduces to 
the identity. For example, 7,=(1+7) SND: 

For the domain R=(1), the group of z?—2=0 is G,; for the domain 
(1, w), w being an imaginary cube root of unity, the group is the cyclic group 
C, (Exercises 3 and 6, page 58). Call the roots 


4,=8/2, 2%, =w8/2=02,, 2%, =w*/2=wx,. 
Then w=2,/z,, a rational function belonging to C;. In fact, (x,x.%3) replaces 
X,/t, by X/%,=W=7,/%X,, (2X37) replaces x,/z, by 2,/%;—G =a 
these two substitutions and the identity are the only substitutions leaving 
z,/t, unaltered. If we subsequently adjoin ¥/2, the roots all belong to the 
enlarged domain (1, w, ¢/2), so that the group reduces to the identity. 


Src, 74] THEORY OF ALGEBRAIC EQUATIONS. 63 


74. In general, we are given a domain R=(R’, R”,...) and 
an equation /(z)=0 with coefficients in that domain. Let G be 
its group for R. Adjoin a quantity €. The irreducible Galois 
resolvent F’,(V)=0 for the initial domain R may become reducible 
in the enlarged domain R,=(¢; R’, R”’,...). Let a(V, §&) be 
that factor of F,(V) which is rational and irreducible in R, and 
vanishes for V=V,. If V,, Va,..., V,are the roots of A(V, €)=0, 
then G’= {I,a,...,k} is the group of f(x) =0 in R, (§ 57). Hence 
G’ is a subgroup of G, including the possibility G’=G, which occurs 
if F,(V) remains irreducible after the adjunction of &, so that 
AV, €)=F(V). 

THEOREM. By an adjunction, the group G is reduced to a sub- 
group G’. 

75. Suppose that, as in the examples in § 73, the quantity 
adjoined to the given domain RF is a rational function ¢(x,, 2%, ...,%n) 
of the roots with coefficients in R. 

TuHrorEeM. By the adjunction of a rational function $(a,,...,%n) 
belonging to a subgroup H of G, the group G of the equation is reduced 
precisely to the subgroup H. 

It is to be shown that the group H has the two characteristic 
properties (§ 61) of the group of the equation for the new domain 

ieee ,...). Hirst, any rational function (2), ...,2n) 
which remains unaltered by all the substitutions of H is a rational 
function of ¢ with coefficients in R (§ 72) and hence lies in Ay. 
Second, any rational function f(a, ..., YZ») which equals a quantity 
p in &, remains unaltered by all the substitutions of H. For the 
relation ¢=o may be expressed as a rational relation in & and 
hence leads to a true relation when operated upon by any sub- 
stitution of G (§ 62) and, in particular, by the substitutions of 
the subgroup H. The latter leave ¢, and hence also p, unaltered. 
Hence the left member ¢ of the relation remains unaltered rela all 
the substitutions of H. 


CHAPTER VII. 


SOLUTION BY MEANS OF RESOLVENT EQUATIONS, 


76. Before developing the theory further, it is desirable to 
obtain a preview of the applications to be made to the solution 
of any given equation /(x)=0. Suppose that we are able to solve 
the resolvent equation (15), one of whose roots is the rational 
function ¢ belonging to the subgroup H of the group G of f(x) =0. 
Since ¢ is then known, it may be adjoined to the given domain 
of rationality (R’, R’”,...). For the enlarged domain Aj= 
(J; R’, R”,...), the group of j(7)=0 is H. _ Let yan 
be a rational function with coefficients in R, which belongs to a 
subgroup K of H. Suppose that we are able to solve the resolvent 
equation one of whose roots is y. Then y may be adjoined to the 
domain R,. For the enlarged domain R,=(y, ¢; R’, R”’,...), the 
group of j(z)=0 is K. Proceeding in this way, we reach a final 
domain FR; for which the group of /(x)=0 is the identity G,. Then 
the roots 2,,...,%n, being unaltered by the identity, lie in this 
domain R; (property A, §61). The solution of /(x)=0 may there- 
fore be accomplished if all the resolvent equations can be solved. 
To apply Galois’ methods to the solution of each resolvent, the 
first step is to find its group for the corresponding domain of 
rationality. 

77. Isomorphism. Let G be the group of a given equation 
f(x)=0 for a given domain Rk. Let ¢(x,,...,%,) be a rational 
function of its roots with coefficients in RF and let ¢ belong to a 
subgroup H of index v under G. Consider a rectangular array 

04 


Src. 77) THEORY OF ALGEBRAIC EQUATIONS. 65 


of the substitutions of G with those of H in the first row, and the 
resulting functions conjugate to ¢: 


hy=L ch, EP d= 
92 hog, «+ hpg, f= “Po, 


qv (xt ae : hpg, e. 
Apply any substitution g of the group G to the v conjugates 


(16) $1 Par Pog +++s Pore 

The resulting functions 

(17) $9, Yo,9) Poqus age? E Yo,0 

are merely a permutation of the functions (16), as shown in § 29, 
hence to any substitution g of the group G on the letters x,,..., 2p, 


there corresponds one definite substitution 


r=(5 ae Ho) (‘2 ) 
$y Yong =e Yo,9 Yoi9 
on the letters (16). We therefore obtain * a set J’ of substitutions 
y, not all of which are distinct in certain cases (xs. 2 and 3 below). 


TurorEM. The set I’ of substitutions 7 jorms a group. 
For to g, g’, and gg’ correspond respectively 


r= ($2 Hie: = (5% 4} 7 = (Ses 3) 

Yoo)” Yaa Yajao' 

To compute the product 77’, we vary the order cf the letters in the 
first line of 7’ and have 


r-(be,) =i) 2 


Hence if J" contains y and 7’, it contains the product 77’. 
Since J” contains a substitution replacing ¢ by ¢,, for any 
t=1,..., v, the group I’ is transitive (§ 66). 





* For a definition of J’ without using the function ¢, see § 104. 


66 SOLUTION BY MEANS OF RESOLVENT EQUATIONS. [Cu. VII | 


DEFINITIONS. The group J’is said to be isomorphic to G, since 
to every substitution g of G corresponds one substitution 7 of I’, 
and to the product gg’ of any two substitutions of G corresponds 
the product yy’ of the two corresponding substitutions of J”. If, 
inversely, to every substitution of J" corresponds but one substi- 
tution of G, the groups are said to be simply isomorphic;* other- 
wise, multiply isomorphic.* 


EXAMPLE 1. Let G=G,@), H=G,, $=2,+w2,+wx,. Set (compare § 9) 


hi=$, tr=¢a, $:=$o, fu=Pe, Ys=d, Po=Pe- 


Then a=(2,%,2;) replaces ¢, by ¢, =w¢,, and ¢, by ¢,=w¢,. Hence areplaces 
gy by wf, =¢5, $3 by w$,=¢1, $e by wh,=95, $s by w7f,=¢,. Hence to a 
corresponds a=(¢,¢.¢3)(GaPe;). Similarly, we find that to c=(2#,%3) corre- 
sponds 7 =(¢,44)(¢2¢5)(Y¢3¢¢). Hence to b=a? corresponds 6 =a’, to d=a-'ca 
vorresponds 0 =a—'ya, to e=b~'cb corresponds ¢«=/—17. We have therefore 
the following holoedric isomorphism between G and I’: 


if i 

G = (X,X2X3) a=(Pibos) (Yahos) 
b= (242322) B=(Pidshe) Yupse) 
C= (X2X3) r= (Piha) Yobs) (Pao) 
d = (2,23) 0 = (Poe) (YsPs) (Pres) 
€ = (X42) € = ($345) Pipe) Yoh) 


It may be verified directly that to b, d, e correspond ~, 6, ¢, respectively. 
Since I, a, 8, 7, 0, e replace ¢, by ¢,, ¢2, ¢3, Yu, Ys, Ye, respectively, J is tran- 
sitive. . 
EXAMPLE 2. Let G=G,,(, H=G,, ¢=(2,—2,)(a,—2,). Set 
Pi=P, $2=(%1—X3)(%—X), Y3=(%,—X,) (12 —T3). 


We obtain the following meriedric isomorphism between G and I’: 


I, (2X) (g%q), (yy) (TeX), (Hq) (T_T) | I 
(x2030,), (24230), (11423), (2,222) (Yifoys) 
(2224X3), (2,242), (X,1275), (1324) (Yifs¢2) 


The group J’ is transitive since it contains substitutions replacing ¢, by 


fi; ho, or ?s. 


* Other terms are holoedric and meriedric for simple and multiple — 


isomorphism. 


Src. 78} THEORY OF ALGEBRAIC EQUATIONS. 67 


78. Order of the group J’. To find the number of distinct 
substitutions in J’, we seek the conditions under which two sub- 
stitutions 7 and 7’ of /' are identical. Using the notation of § 77, 
the conditions are 


Yo,0=Vo;0 (i=1, 2, en, v), 


if we set g,=I. Applying to this identity the substitution g-1g;~, 
we get i 


$= g,9'9—19,—1- 


Hence g,g’9~'g;-'=h, where h is some substitution leaving ¢ unal- 
tered and hence in the group H. Then 


/ 


gg *=9;_ "hg; Cae eer ae) 


But g;—*hg; belongs to the group H;=g,~'Hg; of the function ¢,; 
(§ 39). Hence g’g~' belongs simultaneously to H,, H,,..., Hy, 
and therefore to their greatest common subgroup J. 

Inversely, any substitution o of J leaves ¢,, ¢.,...,¢, unal- 
tered and hence corresponds to the identity in J’. Then g and 
g’ =og correspond to substitutions 7 and 7’ which are identical. 

If G is of order k and i the greatest common subgroup J of H,, 
He, , 1, 1s of order 7, then I’ 1s of order k/j. 


EXAMPLE 1. For G=G,, H =G,, the order of I is 6 (§ 77, Ex. 1). 

Exampte 2. For G=G‘), H=G, (§ 77, Ex. 2), we have H,=H,=H,, 
since G, is self-conjugate under G,, (§ 41). Hence k=12, 7=4, so that the 
order of J’ is 3. 

EXAMPLE 3. For G=@Q), H, =G,, $ =2,4,+7,%,, we set (§ 29, Ex. 2) 


Py =U 11, +%3%y, Po=XyX3tXyX, y= 2XyX_tXyXp. 


hen f,—G,, H,—G,, H,=—G,, J=G, (§ 21). Hence I is of order 24=6. 
This result may be verified directly. There are only 6 possible substitutions 
on 3 letters ¢,, %, ¢;. But the substitutions of G which lead to the identical 
substitution of J’ must leave ¢,, ¢,, ¢, all unaltered and hence belong to the 
greatest common subgroups G, of H,, H,, H;. Hence exactly four substitu- 
tions of G correspond to each substitution of J",so that the order of I’ is 24 =6, 
The four substitutions of any set form one row of the rectangular array for 


68 SOLUTION BY MEANS OF RESOLVENT EQUATIONS. [Cu. VIL 


G,, with the substitutions I, (x,%2)(%3%4), (aX 3) (a%,), (184) (as) of G, in the 
first row. As right-hand Pe we a take 


G1=1, g2=(L2%324), = (20423), = (324),  J5=(X2%q), Jo =(L2%s). 
To the Foe substitutions of the first a the four of the second row,..., 
correspond 


I, (diheds), (Yidsh2), (Pods), (dis), (Yi). 


79. Of special importance is the case in which H,, H,,..., Hy 
are identical, so that H 7s self-conjugate under G. Then J=H, 
so that the order k/7 of I’ equals the index »v of H under G. Hence 
the number of distinct substitutions of J” equals the number of 
letters ¢,,...,¢. upon which its substitutions operate, or the 
order and the degree of the group I’ are equal. Moreover, I" was 
seen to be transitive. Hence J" isa regular group (§ 67). 

DEFINITION.* When H is self-conjugate under G, the group I’ 
is called the quotient-group of G by H and designated G/H. In 
particular, the order of G/H is the quotient of the order of G by 
that of H. 


EXAMPLE 1. By Examples 1 and 2 of § 77, the quotient-group G,/G, is 
a regular group on six letters; the quotient-group G,,/G, is the cycle group 
{L, (Gidrs), (YiPs¢2)}, which is a regular group. 

EXAMPLE 2. We may not employ the symbol G,,/G;, since G, is not 
self-conjugate under G,, (§ 78, Ex. 3). 

EXAMPLE 3. Consider the groups G, and G, on three letters. To G, 
belongs ¢, =(a,—2,)(@,—23)(«;—2,); under G, it takes a second value ¢, = —¢, 
($9). We obtain the following isomorphism between G, and I’: 

if (X15), (X,%32) | L 
(X25), (X23), (X4X2), ($2) 
Since G; is self-conjugate under Gs, we have I'=G,/G,={I, (¢,¢,)}. 


CoroLtuary. J] H is a selj-conjugate subgroup of G of prime 
index v, then I’ 1s a cyclic group of order v (§ 27). 

Illustrations are afforded by the groups G,,/G, and G,/G, of Exs. 1 and 2. 

Remark. Any substitution group G is simply isomorphic with 
a regular group. In proof, we have merely to take as ¢ any n!- 


valued function V,, whence I’ will be of order equal to the order 
of G. 











* Holder, Math, Ann., vol. 24, page 31. 


Src. 80] THEORY OF ALGEBRAIC EQUATIONS. 69 


80. Let H be a maximal self-conjugate subgroup of G (§ 48). 
The quotient-group ['=G/H is then simple (§ 43). For if I’ has 
a self-conjugate subgroup 4 distinct from both J’ and the identity 
G,, there would exist, in view of the correspondence between G 
and I’, a self-conjugate subgroup D of G, such that D contains 
H but is distinct from both G and H. This would contradict the 
hypothesis that H was maximal. 

For example, if H is a self-conjugate subgroup of G of prime index yr, 


it is necessarily maximal. Then I’ is a cyclic group of prime order v (Cor., 
§ 79) and consequently a simple group. 


81. The importance of the preceding investigation of the group 


I of substitutions on the letters ¢,, ¢., . . .,¢, lies in the significance 
of I’ in the study of the resolvent equation 
(15) g(y)=(y—P)Y—$2) -- - (Y— Gv) =9, 


whose coefficients belong to the given domain R. We proceed 
to prove the 
THroreM. lor the domain R, the growp of the equation (15) is I’. 
We show that J’ has the characteristic properties A and B of 


§61. Any rational function p(¢,, ¢,...,¢.) with coefficients 
in R may be expressed as a rational function r(x, %;...,2%n) 
with coefficients in R: 

(18) oY) Qo; cy yj =1(2y, Voy eee) In). 


From this rational relation we obtain a true relation (§$ 62) upon 
applying any substitution g of the group G on a,,...,%. But g 
gives rise to a substitution 7 of the group J'on ¢,,...,¢,. Hence 
the resulting relation is 


(19) Or(Pry Poy + + « : Py) =P (2, La). +) En). 


To prove A, let o(¢,,...,¢.) remain unaltered by all the sub- 
stitutions of I’, so that o,=, for any y in I’. Then, by (18) and 
(19), 7,=7, for any ginG. Hence r lies in the domain RF (property 
A for the group G).' Hence a lies in R. 

To prove B, let po lie in the domain R. Then, by (18), r lies 


70 SOLUTION BY MEANS OF RESOLVENT EQUATIONS. [Cu. VII 


in R. Hence rg=r, for any g in G (property B for the group G). 
Hence, by (18) and (19), o,=, so that o remains unaltered by all 
the substitutions 7 of I’. 

Cor. 1. Since J’ is transitive ($ 77), equation (15) is irreducible 
in R (§ 68). This was shown otherwise in § 71. 

Cor. 2. If the group H to which ¢ belongs is self-conjugate 
under G, the group of the resolvent (15) is regular (§ 79). The 
resolvent is then said to be a regular equation. 

Cor. 3. If H is a self-conjugate subgroup of G of prime index », 
the group of (15) is cyclic (§ 79, Corollary). The resolvent is then 
said to be a cyclic equation of prime degree »v. 

Cor. 4. If H is a maximal self-conjugate subgroup of G, the 
group of (15) is simple (§ 80). The resolvent is then said to be 
a regular and simple equation. 

82. THEOREM. The solution of any given equation can be reduced 
to the solution of a chain of simple regular equations. 

Let Gbe the group of the given equation for a given domain R, 
and let a series of composition (§ 48) of G be 


GH eae as 


the factors of composition being A (index of H under G), » (index 
of K under H),...,o Gndex of G, under M). Let @)0 ewe 
be rational functions of the roots belonging to H, K,...,M, G,, 
respectively (§ 70). Then ¢ is a root of a resolvent equation 
of degree A with coefficients in &, which is a simple regular equation 
(§ 81, Cor.4). By the adjunction of ¢ to the domain R, the 
group G of the equation is reduced to H (§ 75). Then ¢ is a root 
of a simple regular equation of degree » with coefficients in the 
enlarged domain (¢, #). By the adjunction of ¢, the group is 
reduced to K. When, in this way, the group has reduced to the 


identity G,, the roots z,,...,2%p lie in the final domain reached 
(compare § 76). 
In particular, if the factors of composition i, u,...,0 are all 


prime numbers, the resolvent equations are all regular cyclic equations 
of prime degrees (§ 81, Cor. 3). 


Src. 83] THEORY OF ALGEBRAIC EQUATIONS. 71 


83. THEOREM. A cyclic equation of prime degree p ts solvable 
by radicals. 

Let & be a given domain to which belong the coefficients of the 
given equation /(z)=0 with the roots 2,2%,,...,2p -,, and for 
which the group cf /(2) =O is the cyclic group G= {/,s,s?,...,s?—1}, 
where s=(%,2,%,...2»-,). Adjoin to the domain # an imaginary 
pth root of unity * w and let the group of /(z)=0 for the enlarged 
domain R’ he G’. Consider the rational functions, with coefficients 
ic.1t;, 

(20) G,= 2) +02, +0%2,+ ... tw Me, _,. 


Under the substitution s, #; is changed into w~6;. Hence 67 =6; 
is unaltered by s and therefore by every substitution of G and of 
the subgroup G’ (§ 74). Hence 9; lies in the domain Ki’ (§ 61). 
Extracting the pth root, we have 0;=~/0,. Since the function (20) 
belongs to tke identity group, it must be possible, by Lagrange’s 
Theorem (§ 72), to express the roots 2%,2,,...,%p-, rationally 
in terms of J;. The actual expressions for the roots were found 
in the following elegant way by Lagrange. We have, by (20), 


Ly tXy +2, Hictas ert yess C 

= wary 
Qytwr, +72, ave Piha eye NG, 

9 ah i hey 
Gytwe, tots, +... +w%P-32,_,—V90, 

é 2 ho 
to twtr, tw De, +... tw? ay = VOp-1 


where c= 6, is the negative of the coefficient of xP—! in f(x) =0. 
Multiplying these equations by 1, w™, w~™,..., w~—*, respect- 
ively, and adding the resulting equations, and then dividing by p, 
we get T 


n= * } ctw V6, +0-#Y/6,+ an + w—P-MN/ Oy 3 l , 
? ) 








* As shown in § 89, w can be determined by a finite number of applications 
of the operation extraction of a single root of a known quantity. 
Poucel toot... +ae—)t=—0 fori=1,2,...,p—1. 


72 SOLUTION BY MEANS OF RESOLVENT EQUATIONS, [Cu. VII 


for i=0, 1,..., p—1. The value of one of these p—1 radicals, 


say / 6,, may be chosen arbitrarily; but the others are then fully 
determined, being rationally expressible in terms of that one. 
Indeed, | 


V6; -(VO,)'=0;+6 


becomes w*0;+(w—10,)* upon applying the substitution s and 
hence is unaltered by s, and is therefore in the domain R’. 


84. From the results of $$ 82-83, we have the following 

THEOREM. I} the group of an equation has a series of composition 
for which the factors of composition are all prime numbers, the equation 
is solvable by radicals, that 1s, by the extraction of roots of known 
quantities. 

The group property thus obtained as a sufficient condition for 
the algebraic solvability of a given equation will be shown (§ 92) 
to be also a necessary condition. 


CHAPTER VIII. 
REGULAR CYCLIC EQUATIONS; ABELIAN EQUATIONS. 


85. Let f(a) =0 be an equation whose group G for a domain R 
consists of the powers of a circular substitution s=(a,x7,... 2p): 


CYR a epics e 


n being any integer. Since the cyclic group G is transitive and of 
order equal to its degree, it is regular (§ 67). Inversely, the gen- 
erator s of a transitive cyclic group is necessarily a circular sub- 
stitution on the n letters.* 

The equation /(z)=0 then has the properties: 

(a) It is irreducible, since its group is transitive (§ 68). 

(b) All the roots are rational functions, with coefficients in R, 
of any one root 2,. Indeed, there are only n substitutions in the 
transitive group on 7 letters, and consequently a single substitution 
(the identity) leaving x, unaltered. Since 2, belongs to the identity 
group, the result follows by Lagrange’s Theorem (§ 72). Let 
xz,=O(x,). To this rational relation we may apply all the substi- 
tutions of G (§ 62). Hence 


(21) %=O(%,), Tz=O(%q), -.-, Ln=O(Xn_1), X= O(In). 


DEFINITION. An irreducible equation for a domain R between 
whose n roots exist relations of the form (21), @ being a rational 
function with coefficients in FR, is called an Abelian equation.t 





* A non-circular substitution, as t=(2x,%,%,)(7,%;), generates an intransi- 
tive group. Thus the powers of ¢ replace x, by 2, x, or x; only. 

t+ More explicitly, uniserial Abelian (einfache Abel’sche, Ironecker). 
A more general type of ‘‘ Abelian equations” was studied by Abel, Gfuvres, 
I, No. XI, pp. 114-140. 


73 


74 REGULAR CYCLIC EQUATIONS ; ABELIAN EQUATIONS. [Cu. VIII 


86. THrorEM. The group G of an Abelian equation is a regular 
cyclic group. 
Denote any substitution of the group G by 


—) [(tpte te Coe eee 
ee Tq Ug Uy ... BM)" 
Applying to the rational relations (21) the substitutions g (§ 62), 


%,=O(Xa); t;=O(x,), ; > 2. =0G eee 


But, by (21), 0(a2)=2a4,, holding also for a=n if we agree to set 
U;=Lian=Tign= .-. Lt follows that 


La= UT a+; XL =Xotay «sens Ta=Ly44. 


Since the equation is irreducible, its roots are all distinct. Hence, 
aside from multiples of n, 


B=atl1, p=Pt+1l=a+2, d=7+1=a43,... 
ae é Sy o5 oT een ) 
-g Vig Lat, Bare! wi alelnn ae 
Since g replaces x; by 2;144_,, it is the power a—1 of the circular 
substitution s=(2,%1%3..¥%n) which replaces x; by 2;,,. Hence 


G is a subgroup of G’={/, s, s?,..., s”-*}. But Gis transitive, 
since the equation is irreducible. Hence G=G’. 


Jal 
EXAMPLE. The equation 7*+2?+2?+2+4+1 = =0 has the roots 


Yy=&, =e, wz=e*, m=, 


where ¢ is an imaginary fifth root of unity. Hence 
T.=%y?, %3=X,?, Ly=X;”, 1 =2,?. 
Moreover, the equation is irreducible in the domain F of all rational numbers 
(§ 88). This may be verified directly by observing that the linear factors 
are «—e? and hence irrational, while 
ett+a3+o?4+e4+1=(2?+axr+r)(a?+b241-') 
gives at+b=1, ab+r+r—'=1, ar—'+br=1, so that either 
a=4(14V5), b=}1FV5), r=, 
r 1 
pend be 
r+1, (ea a 
Hence the group for F is a cyclic group. Compare Ex. 4, page 58. 


or, a rtrtrt+r+ti1 =o. 


Src. 87) THEORY OF ALGEBRAIC EQUATIONS. 75 
87. Cyclotomic equation for the pth roots of ibis p being 
prime, 
(22) ePTitaP ... +2e+1=0. 
Let ¢ be one root of (22), so that «?=1, «Al. Then 
(23) Smee | See el 


* are all roots of (22) and are all distinct. Hence they furnish 
all the roots of (22). As shown in the Theory of Numbers, there 
exists,* for every prime number p, an integer g such that g”—1is 
divisible by p for m=p—1 but not for a smaller positive integer m. 
Such an integer g is called a primitive root of p. It follows that 
the series of integers 

SS eicnsoreed pemmer 
when divided by p, yield in some order the remainders 

Pee 2 an ay 1 


Hence the roots (23) may be written 


2 2— 
Y= &, L= 89, Le= 9,11, Up_y HEF P 

<a a Et) 0 
Ty= 119, Lg=Xy%,. 6.) Lpy~=Up—2, Ty =Xp_y, 


the last relation following from the definition of g, thus: 
(9? *\9 = eo? * — eltap=e, 


Hence the roots have the property indicated by formule (21). In 
view of the next section, we may therefore state the 

THEOREM. The cyclotomic equation for the imaginary pth roots 
of unity, p being prime, 1s an Abelian equation with respect to the 
domain of all rational numbers. 





* For example, if p=5, we may take g=2, since 
2'—l=1, 2?—1=3, 2'—1=7, 2*—1=15. 


For p=5 the results of this section were found in the example of § 86. 


76 REGULAR CYCLIC EQUATIONS ; ABELIAN EQUATIONS. (Cu. VIII 


88. Irreducibility of the cyclotomic equation (22) in the domain 
R of all rational numbers.* Suppose that 
apt +ap? +... +2+1=4(2)-P(2), 
where ¢ and ¢ are integral functions of degree < p—1 with integral { 
coefficients. Taking x=1, we get 
p= (1) -¢(1). 
Since /: is prime, one of the integral factors, say 4(1), must be +1. 


Since $(7)=0 has at least one root in common with (22), whose 
roots are (23), at least one of the expressions ¢(¢*) is zero. Hence 


(24) $(e)-G(e2)-h(e8) ... $(e?71) <0. 
For any positive integer s less than p, the series 
(25) 6; 628 Uc ee eee 


is identical, apart from the order of the terms, with the series (23). 
For, every number (25) equals a number (23), and the numbers 
(25) are all distinct. In fact, if 
ev8= ef, whence e"¥#=1, (OSr<p, OSiap) 
then (r—t)s, and consequently also r—t, is divisible by 7p, so that 
r=t. Hence (24) holds true when « is replaced by «*. Hence 
f(z) f(a?) ... $(a?-4)=0 
is an equation having all the numbers (23) as roots. Its left mem- 
ber is therefore divisible by v?~'+ ...+2+1, so that 
h(x): f(a”)... h(x?!) = Q(x) -(aP-1+2?-74 ... +241), 
where Q(a) is an integral function with integral coefficients. Set- 
ting r=1, we get 


[60 Po =[t1P EHP: O): 


Since +1 is not divisible by :, the assumption that x?—14+ ... +241 -. 


is reducible in FR leads to a contradiction. 


* The proof is that by Kronecker, Crelle, vol. 29; other proofs have been 
given by Gauss, Eisenstein (Crelle, vol. 39, p. 167), Dedekind (Jordan, 
Traité des substitutions, Nos. 413-414). 

{ If rational, then integral (Weber, Algebra, I, 1895, p. 27). 


Sxc. 89] THEORY OF ALGEBRAIC EQUATIONS. 77 


89. THEoREM. Any Abelian equation is solvable by radicals. 
Let n be the degree of the Abelian equation. By § 86, its 


group G is a regular cyclic group {J, s, s?,..., s®—1} of order 1 
. Set n=p-n’, where pis prime. Set s?=s’. Then the group 
THe Te cee nai 


is a subgroup of G of prime index p. It is self-conjugate, since 


389/492 == g—3g2pg8 — gap — g/a 


by § 138. Hence H may be taken as the second group of a series 
of composition of G. Proceeding with H as we did with G, we 
finally reach the conclusion: 

The factors of composition of a cyclic group of order n are the 
prime jactors of n each repeated as often as it occurs in n. 

In view of the remark at the end of § 82, it now follows that 
any Abelian equation of degree n can be reduced to a chain of Abelian 
equations whose degrees are the prime factors oj n. 

We may now show by induction that every Abelian equation 
of prime degree p is solvable by radicals. We suppose solvable 
all Abelian equations of prime degrees less than a certain prime p. 
Among them are the Abelian equations of prime degrees to which 
can be reduced the Abelian equation of degree p—1, giving an 
imaginary pth root of unity ($87). The latter being therefore 
known, every Abelian equation of degree p:p is solvable by radicals 
(§ 83). Now an Abelian equation of degree 2 is solvable by radicals. 
Hence the induction is complete. 

It follows now that an Abelian equation of any degree is solvable. 

Corotiary. If p is a prime number, all the pth roots of unity 
ean be found by a finite number of applications of the operation 
extraction of a single root of a known quantity, the index of each 
radical being a prime divisor of p—1. 

90. Lemma. Jf p be prime, and if A be a quantity lying in a 
domain R but not the pth power of a quantity in R, then xP?—A 18 
wreducible in R. 

For, if reducible in R, so that 


x —A=¢,(xr)-¢,(x)..., 


78 REGULAR CYCLIC EQUATIONS ; ABELIAN EQUATIONS. (Cu. VIII 


the several factors are of the same degree only when each is of 
degree 1, the only divisor of p. In the latter case, the roots would 
all lie in R, contrary to assumption. Let then ¢, be of higher de- 
gree than ¢, and set 

bi() =(@#— 4)... (4-27), Pole) =(4@—- 2’) . . . (4-2), 
so thatn,—n,>0. The last coefficients in the products are 

105 ie Crs + w2,"1, er ee Xn, = + w2x,"2, 

respectively, since the roots of 7?—A=0 are 
(26) Eas GIL Ge, Wa ee 


w being an imaginary pth root of unity. But the last coefficients, 
and their quotient +w°%x,”, where m=n,—n,>0,liein R. Since 
p and mare relatively prime, integers and v exist for which 


mu—pv=1. 


*. (w°t,™) P= ta Pe = tA a = Ae 


where x’ is one of the roots (26). Hence A,z2’, and consequently 
x’, liesin R. Then A equals the pth power of a quantity z’ in R, 
contrary to assumption. Hence 2?—A must be irreducible. 
91. THEOREM. A binomial equation of prime degree p, 
zP—A=0, 
can be solved by means of a chain of Abelian equations of prime degree. 
Let R be the given domain to which A belongs. Adjoin w 
and denote by R’ the enlarged domain. Then the roots (26) 
satisfy the relations 


of the type (21) of $85, A(x) being here the rational function wa. 
The discussion in § 90 shows that x?—A is either irreducible in the 
enlarged domain f’ or else has all its roots in R’. In the former 
case, the group of x?—A=O for RF’ is a regular cyclic group (§ 86); 
in the latter case, the group for R’ is the identity. But w itself is 
determined by an Abelian equation (§ 87). Hence, in either case, 
xz? —A=0 is made to depend upon a chain of Abelian equations, 
whose degrees may be supposed to be prime (§ 89). 


CHAPTER IX. 
CRITERION FOR ALGEBRAIC SOLVABILITY. 


92. We are now in a position to complete the theory of the 
algebraic solution of an arbitrarily given equation of degree n, 


(1) 2y—u! 


A group property expressing a sufficient condition for the algebraic 
solvability of (1) was established in § 84. To show that this 
property expresses a necessary condition, we begin with a dis- 
cussion of equation (1) under the hypothesis that it is solvable 
by radicals, namely (§ 50), that its roots 2,,...,2, can be derived 
from the initially given quantities R’, R’,... by addition, sub- 
traction, multiplication, division, and extraction of a root of any 
index. These indices may evidently be assumed to be prime 
numbers. If €, 7,..., ¢ denote all the radicals which enter the 
expressions for all the roots 2, 2,,..., %», the solution may be 
exhibited by a chain of binomial equations of prime degree: 


Poh giv, ..), ni=M(e, RR’, ...), «oes 
ge= Pi. oe) I g, Lae Calle a oF 
x,= RY, cee 7) o FR’, R”, a) (i=1, -.+,%), 


L, M,..., P, R; being rational functions with integral coefficients, 
in which some of the arguments €, 7,... written may be wanting. 
By $91, each of these binomial equations, and therefore also the 
complete chain, can be replaced by a chain of Abelian equations 
of prime degrees: 

79 


80 CRITERION FOR ALGEBRAIC SOLVABILITY. (Cu. IX 


Diy hei owe = Os Abelian for domain R; 
WES Od re R”,. . =O, Abelian for (y, ay : 


Abie - ne: ‘bie R’,. eas Abelian fomame we 4 BY: 
a= Od ee Roe (ta Bt. 


We begin by solving the first Abelian equation O(y)=0 and 
adjoining one of its roots, say y, to the original domain R; the 
group G of (1) then reduces to a certain subgroup, say H, 
including the possibility H=G (§ 74). Then we solve the second 
Abelian equation Y(z)=0 and adjoin one of its roots, say z, to the 
enlarged domain (y, #); the group H reduces to a certain sub- 
group, say J, including the possibility J=H. Proceeding in this 
way, until the last equation 0(w)=0 has been solved and one of 
its roots, say w, has been adjoined, we finally reach the domain 
(w,..., 2, y, R), with respect to which the group of (1) is the 
identity G,, since all the roots x; lie in that domain. 

By every one of these successive adjunctions, either the group 
of equation (1) is not reduced at all or else the group is reduced 
to a self-conjugate subgroup of prime index. This theorem, due to 
Galois, is established as a corollary in the next section; its impor- 
tance is better appreciated if we remark that each adjoined 
quantity is not supposed to be a rational function of the roots, in 
contrast with § 75, so that we shall be able to draw an important 
conclusion, due to Abel, concerning the nature of the irrationalities 
occurring in the expressions for the roots of a solvable equation 
($ 94). 

From this theorem of Galois, it follows that the different groups 
through which we pass in the process of successive adjunction 
of a root of each Abelian equation in the chain to which the given 
solvable equation was reduced must form a series of composition 
of the group G of the given equation having only prime numbers 
as factors of composition. Indeed, the series of groups beginning 
with G and ending with the identity G, are such that each is a self- 
conjugate subgroup of prime index under the preceding. Hence 
the sufficient condition (§ 84) for the algebraic solvability of a 


Src. 93] THEORY OF ALGEBRAIC EQUATIONS. 81 


given equation is also a necessary condition, so that we obtain 
Galois’ criterion for algebraic solvability: 

In order that an equation be solvable by radicals, it is necessary 
and sufficient that its group have a series of composition in which 
the factors of composition are all prime numbers. 

93. Theorem of Jordan,* as amplified and proved hy Holder: + 

For a given domain R let the group G, of an equation F(x) =0 
be reduced to G,’ by the adjunction oj all the roots of a second equation 
F(x)=0, and let the group G, of the second equation be reduced to 
G./ by the adjunction of all the roots of the first equation I’,(x)=0. 
Then G,{ and G,! are self-conjugate subgroups of G, and G, re- 
spectively, and the quotient-groups G,/G,' and G,/G,/ are simply 
asomor phic. 

Let ¢,(&, €,..-, €n) be a rational function, with coefficients 
in R, of the roots of the first equation which belongs to the sub- 
group G,’ of the group G;, of the first equation (§ 70). By hypothe- 
sis, the adjunction of the roots 7,, 7.,...,%m of the equation 
F(x) =0 reduces the group G, to G,’.. Hence ¢, lies in the enlarged 
domain, so that 


(27) PSs, Sar - +25 End=PilMy Nay -- +> Nm)s 


the coefficients of the rational function ¢, being in R. 

Let ¢,, ¢.,..., ¢%~ denote all the numerically distinct values 
which ¢, can take under the substitutions (on €,,..., &n) of G. 
Then G,’ is of index k under G, ($71). Let ¢,, d,..., 6; denote 
‘all the numerically distinct values which ¢, can take under the 
substitutions (on 7,,...,%m) of G,. The k quantities ¢ are the 
roots of an irreducible equation in R (§ 71); likewise for the / 
quantities ¢. Since these two irreducible equations have a com- 
mon root ¢, = ¢,, they are identical (§ 55, Cor. II). Hence ¢,,..., dx 
coincide in some order with ¢,,..., $7; in particular, k=l. 

If s; is a substitution of G, which replaces ¢, by its conjugate ¢,, 
then s; transforms G,’, the group of ¢, by definition, into the group 
of ¢; of the same order as G,’. But ¢;, being equal to a ¢d, lies in 





* Traité des substitutions, pp. 269, 270. + Math. Annalen, vol. 34. 


82 CRITERION FOR ALGEBRAIC SOLVABILITY. [Cu. IX 


the domain R’=(R; 7,, ..., Nm), and hence is unaltered by the 
substitutions of the group G,/ of the equation /’,(~#)=0 for that 
domain R’ (§ 61, property B). Hence the group of ¢; contains 
all the substitutions of G,’; being of the same order, the group 
of ¢; 1s identical with G,’. Hence G,’ is selj-conjugate under G,. 
The group of the irreducible equation satisfied by ¢, is therefore 
the quotient-group G,/G,’ (§ 79). 

Let H, be the subgroup of G, to which belongs ¢,(,, qo, . . - ; Nm)- 
Since ¢, is a root of an irreducible equation in F of degree 1=k, 
the group H, is of index k under G, (§ 71). By the adjunction of 
¢d, (or, what amounts to the same thing in view of (27), by the 
adjunction of ¢,), the group G, of equation /’,(~) =0 for R is reduced 
to H, (§ 75). If not merely ¢,(€,,...,-&,), but all the €’s them- 
selves be adjoined, the group G, reduces perhaps further to a 
subgroup of H,. Hence G,’ is contained in H,. We thus have 
the preliminary result: If the group of F',(2)=0 reduces to a 
subgroup of index k on adjoining all the roots of F',(7)=0, then 
the group of F,(x)=0 reduces to a subgroup of index k,, k,=k, 
on adjoining all the roots of F(x) =0. 

Interchanging /’, and F’, in the preceding statement we obtain 
the result: If the group of /’,(z)=0 reduces to a subgroup of 
index k, on adjoining all the roots of /’,(2)=0, then the group of 
F,(x)=0 reduces to a subgroup of index k,, k,>k,, on adjoining 
all the roots of F,(%)=0. Since the hypothesis for the second 
statement is identical with the conclusion for the first statement, 
it follows that 


k,=k, k, Sk, ky ky, 


so that k,=k. Hence the group G,/ of the theorem is identical 
with the group H, of all the substitutions in G, which leave ¢, 
unaltered. It follows that G,’ 7s self-conjugate under G, (for the 
same reason that G,’ is self-conjugate under G',). The irreducible 
equation in Ff satisfied by ¢, has for its group the quotient-group 
G,/G,!. 

But the two irreducible equations for FR satisfied by ¢, and ¥,, 
respectively, were shown to be identical. Hence the groups 


SEc. 94] THEORY OF ALGEBRAIC EQUATIONS. 83 


G,/G, and G./G,/ differ only in the notations employed for the 
letters on which they operate, and hence are simply isomorphic. 

CorouuARY. For the particular case in which the second equa- 
tion is an Abelian equation of prime degree p, all of its roots are 
rational functions in # of any one root, so that by adjoining one 
we adjoin all its roots. By the adjunction of any one root of an 
Abelian equation of prime degree p, the group of the given equation 
I’ (a)=0 either is not reduced at all or else is reduced to a self- 
conjugate subgroup of index p. 

94. If G, is simple and if the adjunction causes a reduction, 
then G, is reduced to the identity. Hence the group G,’=H),, to 
which belongs ¢,, is the identity. Hence the roots 7,, 7, ..-, Mm 
of /’,(2)=0 are rational functions in FR of ¢, (§ 72) and therefore, 
in view of (27), of the roots £,,..., €n of F,(x)=0. 

Ij the group of an equation F',(x)=0 for a domain R is reduced 
by the adjunction of all the roots of an equation F’,(x) =0 whose group 
for R is simple, then all the roots of F',(x)=0 are rational functions 
im KR of the-roots of F(x) =0. 

Since the group of a solvable equation j(#)=0 has a series of 
composition in which the factors of composition are all prime num- 
bers, the equation can be replaced by a chain of resolvent equations 
cach an Abelian equation of prime degree (end of § 82, § 85). 
The adjunction of a root of each resolvent reduces the group of the 
equation and the group of the resolvent is simple, being cyclic of 
prime order. Hence the roots of each Abelian resolvent equation 
are all rational functions of the roots of f(z)=0. But the radicals 
entering the solution of an Abelian equation of prime degree are 
rationally expressible in terms of its roots and an imaginary pth 
root of unity (§ 83), 


4/0, = 2) tue, +w2,+ Te le ge ae er a 


and hence are rationally expressible in terms of the roots of f(x) =0 
and pth roots of unity. We therefore state Abel’s Theorem: 

The solution of an algebraically solvable equation can always 
be performed by a chain of binomial equations of prime degrees whose 


84 CRITERION FOR ALGEBRAIC SOLVABILITY. (Cu. IX 


roots are rationally expressible in terms of the roots of the given equation 
and of certain roots of unity. 

The roots of an algebraically solvable equation can therefore 
be given a form such that all the radicals entering them are 
rationally expressible in terms of the roots of the equation and of 
certain roots of unity. This result was first shown empirically by 
Lagrange for the general quadratic, cubic, and quartic equations 
(see Chapter I). 

The Theorem of Abel supplies the step needed to complete the 
proof of the impossibility of the algebraic solution of the general 
equation of degree n >4 (§ 48). 

95. By way of illustrating Galois’ theory, we proceed to give 
algebraic solutions of the general equations of the third and fourth 
degrees by chains of Abelian equations. 

For the cubic x?—c,x?+¢,7—c,=0, let the domain of rationality 
be R=(c,, C,, ¢,). The group of the cubic for R is the symmetric 
group G, (§ 64). To the subgroup G, belongs 


A = (1 —X_)(L_—X3)(X3—2,). 
In view of Ex. 3, page 4, 4 is a root of the equation 
(28) A? =c,7c,” + 18¢,¢,C, —4¢,3 — 4c,3c, — 27,7. 


Its second root —Z is rationally expressible in terms of the first 
root 4, and (28) is irreducible since 4 is not in R& for general ¢,, ¢, cs. 
Hence (28) is Abelian ($85). By adjoining 4 to R, the group 
reduces to G, (§ 75). Solve the Abelian equation w?+w+1=0 
(§ 87) and adjoin w to the domain (4, R). To the enlarged domain 
R’=(w, 4, ¢,, CG, ¢3) belong the coefficients of the function 


J, =2,+wx,+ wr. 
By § 34, ¢,? has a value lying in Rf’, namely, 
$F =4[2c,3 —9c,c, + 27c, —3(w—w?) A]. 


This binomial is an Abelian equation for the domain R’ (§ 91). 
By the adjunction of ¢,, the group of the cubic reduces to the 


Sxc. 96] THEORY OF ALGEBRAIC EQUATIONS. 85 


igentbys Hence z,, 2, 7, lie in the domain (¢,, w,'4, ¢,, c, ¢,). 
Thus, by § 34, 


oo oe 
n=t(atht" di, "), Xo =3(at0% to ae). 


1 





We may, however, solve the cubic without adjoining w. In 
the domain (4, ¢,, ¢, ¢,), the cubic itself is an Abelian equation, 
since its group G, is cyclic (§ 85). By the adjunction of a root 2, 
of this Abelian equation, the group reduces to the identity, so that 
%, and x, must lie in the domain (2,, 4, ¢,, ¢&, ¢,). The explicit 
expressions for x, and 2, are given by Serret, Algébre supérieure, 
lew. NO. O11: 


Lo 3 { (6c, —2c,”) x," + (9e,—7¢,C.+2¢,°— 4) x, + 4¢,”—¢,7c,—3¢,C,+¢,4}, 


the value of x, being obtained by changing the sign of 4 throughout. 
96. For the general quartic x*+a2°+ bx?+cx+d=0, the group 
for the domain R=(a, b, c, d) is G.,. To the subgroup G,, belongs 
= (X, —Xp)(1, — Lg) (Ly — Ly) (Lp — L3) (Lo — Lq) (Lg — 4). 
Since 4? is an integral function of a, b, c, d with rational coefficients 
(§ 42), we obtain 4 by solving an equation which is Abelian for R. 
After the adjunction of 4, the group is G,. To the subgroup G, 


cf G,. belongs the function y,=2,7,+2,2,. It satisfies the cubic 
resolvent equation (§ 4) 


(16) y>— by? + (ac—4d)y—a*d+4bd—c?=0. 


The group of this resolvent for the domain (4, a, b, c, d) is a cyclic 
group of order 3 (§ 79, Cor.), so that the resolvent is Abelian. By 
the adjunction of y,, the group of the quartic reduces to G,. To 
the subgroup G, of G, belongs the function t=a,+2,—2,—a, It 
is determined by the Abelian equation (§ 5) 


(29) Me i= 0745 £47), 


By the adjunction of t, the group reduces to G,. To the identity 
subgroup G, of G, belongs 2,; it is a root of (17), § 4: 


w+ 4(a—t)x+ hy, — (say, —0)/t=0. 


86 CRITERION FOR ALGEBRAIC SOLVABILITY. (Cu. 1X 


After the adjunction of a root x, of this Abelian equation, the 
eroup is the identityG,. Hence (§ 72) all the roots lie in the domain 
(x,,t, y,,4,a,6,c,d). This is evident for x,, since 2,+2,= —4(a—2). 
For z, and x,, we have 


Vet Lg=Lj+X,—t, %—2y=(Y2— Ys) + (2, —M), 


while y, and yz are rationally expressible in terms of y,, 4, and 
the coefficients of (16), as shown at the end of § 95. In fact, 
(Y1—Y2)(Yo—Ys)(Yi— Ys) has the value 4 by § 7. 

97. Another method of solving the general quartic was given 
in § 42. For the domain R= (a, a, b, c, d), where w is an imaginary 
cube root of unity, the group is G,, (§ 64). After the adjunction 
of 4, the group is G.. To the self-conjugate subgroup G, belongs 
f=Y, + wy,+wy;, where y,=2,7,+27,2,, etc., so that ¢, is a rational 
function of 2,, 2, 73, %,, with coefficients in R. By § 42, 


$= 3(w—w?) A — 2160, 


so that ¢, 1s determined by an equation which is Abelian for the 
domain (4, w, a, b, c, d). Then, by § 42, y;, y., ys; belong to the 
enlarged domain (¢,, 4, w, a, b, ¢, d). 

By the adjunction of t, a root of the binomial Abelian equa- 
tion (29), the group reduces to G,. By the adjunction * of both 
t=V —1 and V=a2,—2,+i2,—ix,, which is a root of a binomial 
quadratic equation (§ 42), the group reduces to the identity G,. 
The expressions for x,, 2%, 2%, %, In terms of #; V, 1, and a, are 
given by formula (41), in connection with (40), of § 37. 








* Without adjoining 7 and V, we may determine t,=2,+27,—x,—2, from 
t,2=a?—4b+4y,. Then t,=2,+2,—x,—2; is known, since t,t,t;=4ab —8c—a? 
by formula (39) of § 36, where t,=¢. Then 

x,=1(—a+i,+i,4+4,), %,=}(—a+t,—t,—t3), etc. 


CHAPTER X. 
METACYCLIC EQUATIONS; GALOISIAN EQUATIONS. 


98. Analytic representation of substitutions. Given any sub- 
stitution 
ga (To TM 22 +++ ny 
De aly ele ie ie 


so that a,b,..., k form a permutation of 0,1,..., n—1, it is pos 
sible to construct a function (z) of one variable z such that 


fO)=a, P(1)=b, (2=c, ..., s(m—1)=k. 


Indeed, such a function is given by Lagrange’s Interpolation- 
Formula, 
ak (z) bF(z) 


a kF(2) 
#2) = 2870) DED) 


+ eee + (2—n4+ 1)’ (n—1)? 


where F'(z)=2(z—1)(2—2)...(g—n+1) and F’(z) is the deriva- 
tive of /’(z). Then the substitution s is represented analytically 


We confine our attention to the case in which n is a prime num- 
ber p, and agree to take 7,=%,4p=%z4.p= .... » Then (as in § 86) 
the circular substitution t=(a, 7, 2,...2%p_,) May be represented 


in the form 
t ( z ) ° 
Ve+1 


Let G be the largest group of substitutions on a, 2,..., Lips 
87 


88 METACYCLIC AND GALOISIAN EQUATIONS. (Cu. X 


under which the cyclic eroup H={I,t,t?,...,i?—*} is self-conjugate. 
The general substitutions g of G and h of H may be written 


x HY 
ee ie )=e 
J Ss Vz+a 


By hypothesis, g~'tg belongs to H and hence is of the form 2’. 


~1_ ($02) ee) 1 = (740) ih 
: é ), d Tz 41)’ ft! Lp(z+1) 


But t” replaces xyz) by %@)4a- Hence must 


Lp(z +1) = T(z) +a 


Taking in turn z=0, 1, 2,..., and writing 6(0) =), we get 


L(y) =Vb+a, VUd(2)= (1) +a=Vb+2a, Vo(3)= Vb(2)+a = Vh+3ay 


By simple induction, we get 42)=2%p+za for any integer z. Hence 


ly 2D oe) . 


Here a and }=¢(0) are integers. Also a is not divisible by 7, since 
g~tg is not the identity. The distinct substitutions * g are obtained 
by taking the values 

a=1,2,..5 ,p—1;-0=0, 12. ee 


The resulting p(p—1) substitutions form a group called the meta- 
cyclic group of degree p. This follows from its origin or from 


(ia) (eecsg) 7 eee) er 
Laz+b Laz4+ 3 La(az+b)+ B Xaaz+(ab+ f) , 


Remark. The only circular substitutions of period p in the 
metacyclic group are the powers of t. For a=1, (80) becomes ?#; 
for a1, (30) leaves one root unaltered, namely, that one whose — 
index z makes az+b and z differ by a multiple of p. 








* Formula (30) does, indeed, define a substitution on 2X, 2, ..., @n—:, 
(7 Ly ots Danke a ) 
XT VXatb Xoat+b..- Z 
since b, a+b, 2a+b,..., (p—1)a+b give the remainders 0, 1, 2, .., p—l, 


in some order, when divided by p. In proof, the remainders are all different. 


Src. 99] THEORY OF ALGEBRAIC EQUATIONS. 89 


99. A metacyclic equation of degree p is one whose group G 
for a domain Ff is the metacyclic group of degree p. It is irre- 
ducible since G is transitive, its cyclic subgroup H being transitive. 
Again, all its roots are rational functions of two of the roots with 
coefficients in #&. For, by the adjunction of two roots, say zu 
and z,, the group reduces to the identity. Indeed, if g leaves 
tu and x, unaltered, then 


(au+b)—u, (av+b)—v 


are multiples of p, so that their difference (a—1)(uw—v) is a multi- 
ple of p, whence a=1, and therefore b=0. Hence the identity 
alone leaves xu and 2y unaltered. 

DerinitTion. Fora domain &, an irreducible equation of prime 
degree whose roots are all rational functions of two of the roots is 
called a Galoisian equation. 

Hence a metacyclic equation 1s a Galovsian equation. 

100. Given, inversely, a Galoisian equation of prime degree 7, 
we can readily determine its group G for a domain R. The equa- 
tion being irreducible, its group is transitive, so that the order 
of G is divisible by p ($ 67). Hence G contains a cyclic subgroup 
H of order p (see foot-note to § 27). Let 2) and x, denote the two 
roots in terms of which all the roots are supposed to be rationally 
expressible. Among the powers of any circular substitution of 
period p, there is one which replaces x, by x,. Hence, by a suitable 
choice of notation for the remaining roots, we may assume that 
‘H contains the substitution 


aed Ho gt nen eae 
To show that H is self-conjugate under G, it suffices to prove 
that any circular substitution, contained in G, 


r= (x, Xi Li, btu Lin _,) 


is a power of ¢; for, the transform of ¢ by any substitution of G will 
then belong to H ($40). Since every two adjacent letters in r 
are different, ¢,,,—2, 1s never a multiple of p and hence, for at 


go METACYCLIC AND GALOISIAN EQUATIONS. [CH. X 


least two values » and »v of z chosen from the series 0, 1,..., p—1 
gives the same remainder when divided by p. Hence 


y] 


Ue gt Pi, 4 y—tyr BAY, Lk 


Since r is a power of a circular substitution replacing 2, by 2x,, we 
may assume that 7,=0, 7,=1. The hypothesis then gives 


Lj,= 8 al X;,, xi.) (a=0, Ly. eee p=); 


where @, is a rational function with coefficients in R. Applying 


to these rational relations the substitutions r“¢ ““ and r’t ™ of the 
group G, we obtain, by § 62, 


Diy yin =F alXo, Lk), Vig, ,-i,=FulXq, Le). 
Hence the subscripts in the left members are equal, so that 
Vasp—laty=ty—=C (das L, wis De 


omitting multiples of p. Hence every subscript in r exceeds by c 
the (u—v)th subscript preceding it. Hence r is a power of ¢. 

Since G has a self-conjugate cyclic subgroup H, it is contained 
in the metacyclic group of degree p (§ 98). | 

The group of a Galovsian equation of prime degree pis a subgroup 
of the metacyclic group of degree p. 

101. A metacyclic equation is readily solved by means of a 
chain of two Abelian equations. Let $= R(x, 2,,..., %p_,) belong 
to the subgroup H of G.* Then 


J=¥¢, J.= R(X, Vo, XL4y vee) Lop 2) a Pp_.= R(X, Up—1) Lop —2) <s'59 X(p—1)?) 


are the p—1 values of ¢ under G. But ¢, is changed into ¢y,; by 
the substitution which replaces x, by a;z,. It follows that the 
p—1 values of ¢ are permuted cyclically under the p(p—1) sub- 
stitutions of G. The group of the resolvent equation 


(w—,)(w—gy) ... (w—Pp_y) =0 


is therefore a cyclic group of order p—1, so that the resolvent 
is an Abelian equation ($85). By the adjunction of ¢, the group 


Src. 101] THEORY OF ALGEBRAIC EQUATIONS. gi 


of the original equation reduces to the cyclic group H, so that it 
is Abelian in the enlarged domain. 
The method applies also to any Galoisian equation. Its group 
G is a subgroup of the metacyclic group and yet contains H as a 
subgroup. The order of G is therefore pd, where d is a divisor 
of p—1. The two auxiliary Abelian equations are then of degrees 
d and p respectively.. Applying § 89, we have the results: 
A Galoisian equation can be solved by a chain of Abelian equations 
of prime degree and hence 1s solvable by radicals. 
EXAMPLE 1. Let A be a quantity lying in a given domain R& but not 
the pth power of a quantity in R. Then the equation 
xP—A =0 
is irreducible in R (§ 90). Its roots are 
Loy Ly =WXy, Ly=W'Xy, «64, Lp =wP—'N, 
All the roots are rationally expressible in terms of x) and 2: 
Cp= (2) "Lo (1=0,1,..., p—l). 
The equation is therefore a Galoisian equation. For the function ¢ belonging 
to the cyclic subgroup H we may take 
T,X Xo 
ae 7S tp 
The resolvent equation w?—'+...+w+1=0 is indeed Abelian (§ 87), 


After the adjunction of w, x? —A =0 becomes an Abelian equation (§ 91). 
EXAMPLE 2. To solve the quintic equation * 





=W, 


(e) y+ py? +sp'y +r=0, 
set y=z— z. Then (compare the solution of the cubic, § 2) 
5 
o—i-.+7=0, 


Seneca Ou P) ; 
2 8=—StVQ, Q=7+\(z)- 
If ¢ is an imaginary fifth root of unity, the roots of (e) are 


Y,=A+B, y=cA+eB, yy=P’A+eB, y=PA+eB, y,=e'd+cB, 
where 


r ~ r = 





* Compare Dickson’s College Algebra, pages 189 and 193. 


92 METACYCLIC AND GALOISIAN EQUATIONS. (Cu. X 


Evidently A and B may be expressed as linear functions of y, and y,. Hence 
Y3, Yas Ys are rational functions of y, and y, with coefficients in the domain 
R=(e, p, r). For general p and r, equation (e) is irreducible in R, since no 
one of its roots lies in # and since it has no quadratic factor in & (as may be 
shown from the form of the roots). Hence (e) is a Galoisian equation. 

102. Lemma. Jf L is a_self-conjugate subgroup of K of prime 
index v and ij k 1s any substitution of K not contained in L, then k”, 
and no lower power of -k, belongs to L, and the period of k 1s divisible 
by v. 

By the Corollary of § 79, the quotient-group K/L is a cyclic 
group 


ef 


La oe ee 


Hence to k corresponds a power of 7, say 7*, where « is not divisible 
by v. Then to k” corresponds (7*)”=J, so that k” belongs to L. 
If 0<m<v, k™ does not belong to L, since (7*)”=TI requires that 
xm be divisible by the prime number »v. 

Let the period yu of k be written in the form 


e=qu+t (OSt<y). 

Since k”=h, a substitution of L, we get [=k*=h*%k". Hence © 

k7=h~4, so that t=0, in view of the earlier result concerning 
powers of k. Hence y is divisible by »v. 

103. THEOREM (Galois). Every irreducible equation of prime degree 


p which is solvable by radicals is a Galoisian equation. 
Let G be the group of the equation for a domain R and let 


(31) Gen a AAG eee 


be a series of composition of G. Since the equation is solvable by 
radicals, the factors of composition are all prime numbers (§ 92). 
Since the equation is irreducible in FR, G is transitive (§ 68), so that 
its order is divisible by p (§ 67). Hence (foot-note to § 27), G 
contains a circular substitution of period p, say t=(% 2... Lp_4). 
Let K denote the last group in the series (81) which contains t. 
Then the group L, immediately following K, and of prime index v 
under K, does not contain t. Since t’=/J belongs to L, while no 
lower power of ¢ belongs to L, it follows from § 102 that v=p. 


Src. 102] THEORY OF ALGEBRAIC EQUATIONS. 93 


To show that L is the identity G,, suppose that LZ contains a 
substitution s replacing x, by a different letter x3. Then w=st+~é 
leaves x, unaltered and belongs to K. Since a—@ is not divisible 
by p and since ¢ does not belong to L, it follows that uw does not 
belong to LZ. By the Lemma of § 102, the period of w is divisible 
by v=p. ‘This is impossible since wu is a substitution on p letters, 
one of which remains unaltered. | 

Since L=G, and the index of Z under K is p, the group K is 
the cyclic group of order p formed by the powers of t. Since the 
group J immediately preceding K in the series (31) contains the 
cyclic group K as a self-conjugate subgroup, J is contained in 
the metacyclic group of degree p (§ 98). By the remark at the 
end of § 98, J contains no circular substitutions of period p other 
than the powers of t. If J’ be the group immediately preceding 
J in the series (31), so that J is self-conjugate under J’, the trans- 
form of ¢ by any substitution of J’ belongs to J and is a circular 
substitution of period p, and therefore is a power of t. Hence the 
cyclic group K is self-conjugate under J’, as well as under J. 
Hence J’ is contained in the metacyclic group (§$ 98). Proceeding 
in this way until we reach the group G, we find that G is contained 
in the metacyclic group. The theorem therefore follows from § 101. 


CHAPTER Af. 
AN ACCOUNT OF MORE TECHNICAL RESULTS, 


104. Second definition of the group J’ of § 77. To show that 
T is completely defined by the given groups G and H and is entirely 
independent of the function ¢ used in defining it, we define a group 
I’, independently of functions belonging to H and prove that 
deol ad 

Consider a rectangular array of the substitutions of G with 
those of the subgroup H in the first row: 


1,10) =) Sie hy 
(32) Tie. (SOS nehe! Pas 

TA ys [isda iste 
where 7; denotes the jth row of the array. Let g be any substitu- 
tion of G. Since g,g,..-, gg lie in the array (32), we may write 
(33) H9=haGar GG=hggs, +++y GoJ=hnGJe- 


Hence the products of the substitutions in the array (32) by g 
on the right-hand may be written (retaining the same order): 


la h a’Q a (hh WY a see (hh a )Ja 
(34) re\he'ga (hohs')gp ... (ash 89 B 
Tk he Gre (hohe) Gr cee (hihe) 9x 


Now ha’, Iola, «++, hd form a permutation of hj=J) i, 

Hence the substitutions in the first row of (84) are identical, apart 

from their order, with those of the ath row of (32). Similarly 
94 


Src. 105] THEORY OF ALGEBRAIC EQUATIONS. 95 


for the other rows. Hence the multiplication of (32) on the right 
by g gives rise to the following permutation of the rows: 


OUe Wate < ae 
iy ne nes a) 
To identify the group I’, of these substitutions 7 with the group 


I’ given by the earlier definition, we note that to g corresponds, 
under the earlier definition, , 


Petes ba, \ bn, ba, be, 
Yo9 Yong ++ Poyg Yo Yo x ++ Pox)’ 
since, by (33), fg.g=%h waa Yay ete. But this substitution differs 


from y only in notation. Hence [’;=I’. 


EXAMPLE 1. Let G be the cyclic group {/, c, c’, c®, c*, c5}, where c°=I, 
and let H be the subgroup jJ, c?}. The array is 


a | Le 
fyeih Cle, Co 

re eet 
TG 





To c corresponds (r,7,7;). Hence I’={J, (ryrers), (ry1's2).}- 

EXAMPLE 2. Let G be the alternating group GS) and let H be the com- 
mutative subgroup G, (§ 21, Ex. /). The rectangular array for G is given 
in § 77, Ex. 2. Multiplying its substitutions on the right by (a,x,)(x%,), 
we obtain the array 


(2,2) (X3%,), I. (112,) (X23); (2,X3) (2_%,) 
(2 ,2%,), (212423), (2,37), (2_23X,) 
(2,227), ‘ (2,23%,), (XptyXs) (24X47) 


Hence each row as a whole remains unaltered, so that to (x,%,)(#3%,) corre- 
sponds the identity. <A like result follows for (x,7)(x,2,) and for the product 
(2,2,)(x,%3) of the two. But (2z,2;7,) applied as a right-hand multiplier 
gives rise to the permutation (7,r,r;) of the rows, as follows immediately from 
the formation of the rectangular array by means of the right-hand multipliers 
pean 42,7,2,)". Hence I’={I, (rirz,), (rirsts }- 


105. Constancy of the factors of composition. By the criterion 
of § 92, an equation is solvable by radicals if, and only if, the 
group G of the equation has a series of composition in which the 
factors of composition are all prime numbers. In applying the 


96 AN ACCOUNT OF MORE TECHNICAL RESULTS. {[Cu. XI 


criterion, it might be necessary to investigate all the series of com- 
positions of G to decide whether or not there is one series with 
the factors of composition all prime. The practical value of the 
criterion is greatly enhanced by the theorem of C. Jordan: * 

If a group has two different series oj composition, the factors of 
composition for one series are the same, apart from thewr order, as 
the factors of composition for the other series. 

EXAMPLE 1. Let G,, G,, H, be defined asin § 21; G,, G), Gy as in Example 
3 of § 65; and let 
Cy={L, (xy grey), (24%) (3%), (210 4%_%3)}, H,={I, (%%2)}, H2= tl, Cree) }. 
Then G, has the following series of compositions: 

Gs, Gy, Gr, G3 Gs, Ga, G2, G3 Gg, Gy, G2, G5 
Gs, Cy, Ga, G13 Gs, Ha, Ga, G3 Gg, Ha, Hr, G,; Gs, Hy, Ho, Gy. 
In each case the factors of composition are 2, 2, 2. 
EXAMPLE 2. Let C,, be the cyclic group formed by the powers of the 


circular substitution @=(2,%%3...2%.). Its subgroups are 
C.= rs a’, a*, a’, a’, a*%}, C,= hg a’, a’, a®}, 
C3= (J, a‘, a*}, C,=(, a}, Ci = {I}. 


The only series of composition of C,, are the following: + 
Cr, Co, Cs, C13 Cra, Co, C2, C13, Cra, Cr, C2, Ch. 
The factors of composition are respectively 2, 2, 3; 2, 3, 2; 3, 2, 2. 


106. Constancy of the factor-groups. In a series of composi- 
tion of G, 
aaG se Wt Arata 


each group is a maximal self-conjugate subgroup of the preceding 
eroup (§ 43). The succession of quotient-groups 


G/G’, GQ’ /G”, G’'/Q/", a. 


forms a series of factor-groups of G. Lach factor-group is simple 
($80). The theorem of Jordan on the constancy of the numerical 





* Traité des substitutions, pp. 42-48. For a shorter proof, see Netto-Cole, 
Theory of Substitutions, pp. 97-100. 
+ Every subgroup is self-conjugate since a—‘aJa'=a/ ($13), 


Src. 107] THEORY OF ALGEBRAIC EQUATIONS. 97 


factors of composition is included in the following theorem of 
Holder :* 

For two_series of composition of a group, the factor-growps of one 
series are identical, apart from their order, with the factor-growps of 
the other series. 


Thus, in Example 1 of § 105, the factor-groups are all cyclie groups of 
order 2. In Example 2, the factor-groups for the respective series are 


kK), k,, K3; K;, K3, Kk; K;, k,, k,, 


where K, and K, are cyclic groups of orders 2 and 3 respectively. That 
C,/C, is the cyclic group K, follows from § 104, Ex. 1, by setting a?=e. 
That C,,./C, is K; follows readily from § 104. 


107. Holder’s investigation f on the reduction of an arbitrary 
equation to a chain of auxiliary equations is one of the most im- 
portant of the recent contributions to Galois’ theory. The earlier 
restriction to algebraically solvable equations is now removed. 
As shown in § 82, the solution of a given equation can be reduced 
to the solution of a chain of simple regular equations by employing 
rational functions of the roots of the given equation. The groups 
of the auxiliary equations are the simple factor-groups G of the 
given equation. Can any one of these simple groups be avoided 
by employing . accessory <rrationalities, namely, quantities not 
rational functions of the roots of the given equation? That this 
question is to be answered in the negative is shown by Holder’s 
result that the factor-groups of G must occur among the groups 
of the auxiliary simple equations however the latter be chosen. 
Any auxiliary compound may first be replaced by a chain of 
equivalent simple equations. The number of factor-groups of G 
therefore gives the minimum number of necessary auxiliary simple 
equations. If this minimum number is not exceeded, then Holder’s 
theorem states that all the roots of all the auxiliary equations are 


* Holder, Math. Ann., vol. 34, p. 37; Burnside, The Theory of Groups, 
p. 118; Pierpont, Galois’ Theory of Algebraic Equations, Annals of Math., 
1900, p. 51. 

+M athematische Annalen, vol. 34, p. 26; Biernonit: Uli, De, 02, 


98 AN ACCOUNT OF MORE TECHNICAL RESULTS» [CH 7x1 


rational functions of the roots of the given equation and the quan- 
tities in the given domain of rationality. 

Holder’s proof of these results, depending of course upon the 
constancy of the factor-groups of G, is based upon the fundamental 
theorem of § 93. 

The special importance thus attached to simple groups has 
led to numerous investigations of them. Several infinite systems 
of simple groups have been found and a table of the known simple 
groups 0: composite orders less than one million has been prepared.* 

For full references and for further developments of Galois’ 
theory, the reader may consult Encyklopadie der Mathematischen 
Wissenschajten, I, pp. 480-520. 





* Dickson, Linear Groups, pp. 307-310, Leipzig, 1901. 


APPENDIX. 


RELATIONS BETWEEN THE ROOTS AND COEFFICIENTS OF AN 
EQUATION. 


Let 2,, %, ..., 2, denote the roots of an equation /(z)=0 in 
which the coefficient of 2” has been made unity by division. Then 


[(«)=(4@—2,)(4— 4)... (@—2Xn), 


as shown in elementary algebra by means of the factor theorem. 
Writing f(x) in full, and expanding the second member, we get 


a — Cae * + cae *— 2... +(—1) "ep Sa" — (a, + 2,+ 2. £2) 2" 
ol Prty te Melet dpe Toles on iy et 
ee ipa Gano Lge ge Pe 


Equating coefficients of like powers of x, we get 
(1) ects ba 9.2 +in=C,, U,2_+ ee Dn a Cos 6 oy L120 + In=Cn. 


These combinations of 2,,...,2, are called the elementary sym- 
metric functions of the roots. Compare Exs. 5 and 6 of page 4. 


FUNDAMENTAL THEOREM ON SYMMETRIC FUNCTIONS.* 


Any integral symmetric function of 1,1, 2,,..., tn can be expressed 
mn one and only one way as an integral function of the elementary 
symmetric functions C,, Co). ~~ 5 Cn 


A term 2x,"17,’"7,"3... is called higher than 2,%7,"27,% ... 
if the first one of the differences m,—n,, m,—Nn., M,—Ns, ..., Which 





* The proof is that by Gauss, Gesammelte Werke, III, pp. 37, 38. 
99 


100 APPENDIX. 


does not vanish, is positive. Then ¢,, ¢, ¢3,..., ¢; have for their 
highest terms @,, 2%», XU, ..-, 2%... 2, Tespeeree ere 
general, the function c,%c,%c,7 ... has for its highest term 


arta desl Be chek ie Aen x,y t a 


Hence it has the same highest term as c,*c,°'c,”. .. if, and only if, 
atB+y+t...=@4+ 8 4+7'+...,Bty+... =P +74 ..5, 
yt... =7'+...; 
whicn require that a=a’, B=[’, y=7’,... | 
Let S be a given symmetric function. Let its highest term be 


h=ax,° Tae LY Le 20 Xp” «a. (aS8sS750 the -5v). 


We build the symmetric function 

C= G0 CPT Gta eee 
In its expansion in terms of 2,,..., 2% by means of formule (1), 
its terms are all of the same degree and the highest term is evi- 


dently h. The difference 
S,=S—o 


is a symmetric function simpler than S, since the highest term h 
has been cancelled. Let the highest term of S, be ) 


h,=a, X4%1 ei U3"1 La eee 
A symmetric function with a still lower highest term is given by 
S,=S,—4, Cagis od Cet ae 


Since the degrees of S, and S, are not greater than the degree of S, 
and since there is only a finite number of terms 2,"%7,"207,%3... 
of a given degree which are lower than the term h, we must ulti- 
mately obtain, by a repetition of the process, the symmetric 
function 0: 

O=S,—ap C,°x—Pr CoP x Ve CaYe—*k oo 


We therefore reach the desired result 


S=a, c,*—8 cf-¥ 22. +, G71 FicPimn .. 7. . «+0, 0,2 Pec Pea Sa 


APPENDIX. Iol 


To show that the expression of a symmetric function S in terms 
of ¢,,..., €n 1S unique, suppose that S can be reduced to both 
P(C,, Co, ---5 Cn) and P(¢,, G, ..., Cn), where ¢ and ¢ are different 
integral functions of ¢,,..., Cn. Then d—d, considered as a func- 
tion of ¢,,..., Cn, is not identically zero. After collecting like 
terms in d—4¢, ‘let bc,*c,%c,¥... be a term with b40. When ex- 
pressed in 2, ..., 2p, it has for its highest term 


b xetBtyt... gBtyt-. gytee,, 


As shown above, a different term b’c,%’c,’c,.” ... has a different 
highest term. Hence of these highest terms one must be higher 
than the others. Since the coefficient of this term is not zero, 
the function d—¢” cannot be identically zero in 2,..., %. This 
contradicts the assumption that S=¢, S=¢, for all values of 
Ge en, Ons 
CoROLLARY. Any integral symmetric function of x, ..., Lyn with 
integral coefficients can be expressed as an integral function of ¢,,..., Cn 
with integral coefficients. 

Examples showing the practical value of the process for the 
computation of symmetric functions are given in Serret, Algébre 
supérieure, fourth or fifth edition, vol. 1, pp. 389-395. 


ON THE GENERAL EQUATION. 


Let the coefficients ¢, ¢,,...,C, be indeterminate quantities. 
The roots 2,, 7,..., 4, are functions of ¢,,..., Cn; the notation 
feet, is definite for each set of values of c,,...,¢,. We 
proceed to prove the theorem: * 

If a rational, integral function of x,,...,% with constant 
coefficients equals zero, it is vdentically zero. 

Peete... %, 1-0. Let.f,,...,.& denote mdeterminates 
and o,,..., 0, their elementary symmetric functions ¢,+...+&n, 


eeeeo es. ..- on. Then 





* This proof by Moore is more explicit than that by Weber, Algebra, II 
(1900), § 566. 


102 APPENDIX. 


MGs, ae s,1= WO. 9 Gal 
the product extending over the n! permutations s,,..., S» of 
1,...,n”, and Y denoting a rational, integral function. Hence 
MPa, , oe) t, |= Pe, yen Cn]=0, 
since one factor ¢[z,;...,%,] is zero... Since se. essere 
indeterminates, W[c,,...,¢n] must be identically zero, i.e., 


formally in ¢,,..., Cn. Consider ¢,,.:., €, tO be sEeeiemneenen 
new indeterminates y,,..., Yn. Then 


PTC oy Yad se ey Onl Uy eee 


formally in y,,..., Yn. Hence, by a change of notation, 
Po,Gy 5% . 7 Sndiy.- 99 Oniein ee 
formally in &,,..., &n. Hence, for some factor, 


blfs)-- +) Sa,)=0 
formally in €,,..., &n. As amere change of notation, 
Oi f. aoa 


As an application, we may make a determination of the group 
of the general equation more in the spirit of the theory of Galois 
than that of § 64. If, in the domain R=(¢,..., Cn), a rational 
function $(2,,...,%n) with coefficients in & has a value lying 
in R, there results a relation 


UA eeGece, in |e 


upon replacing ¢,,..., €n by the elementary symmetric functions 
of %,..-+,%n. By the theorem above, ¢[z,,..., %s,]=0, so that 


Pi, + 94 Le) = Play vy Male 


INDEX. 


(The numbers refer to pages.) 





Abelian equation, 73, 78, 84 Galois’ resolvent, 49, 51 
Abel’s Theorem, 41, 83 | Galoisian equation, 89, 92 
Accessory irrationality, 97 General equation, 30, 40, 55, 101 
Adjunction, 62 . Group, 15 
Alternating function, 18 — of equation, 52, 69, 102 
-— group, 18, 37, 39 
Associative, 11 Holoedric, 66 
Belongs to group, 16, 19, 60 Identical substitution, 10 
Binomial, 31, 34, 41, 78, 91 Index, 21 
Intransitive, 58 
Circular substitution, 13, 37 Invariant, 32 
Commutative, 11, 17 Inverse substitution, 12 
’ Composite group, 37 Irrationality, 1, 4, 8, 84, 97 
Conjugate, 23, 32, 33 Irreducible, 46, 59, 61, 76, 92 
Cube root of unity, 2 Isomorphism, 64, 94 
Cubic, 1, 27, 84 ; 
Cycle, 13 Jordan’s Theorem, 81 
Cyclic group, 16, 68, 70, 74 
Cyclotomie, 75 Lagrange’s Theorem, 24, 61 
Degree of group, 15 Maximal, 36, 69 
Discriminant, 4, 9 Meriedric, 66 
Distinct, 45 Metacyclic, 88, 89 
Domain, 43 Multiplier, 22 
Equal, 45 Odd substitution, 17 
Even substitution, 17, 37 Order of group, 15, 20, 58 
Factor-group, 96 Period, 12, 21 
Factors of composition, 37, 40, 70, | Primitive, root 75 
77, 93, 96 | Product, 11 


103 


104 INDEX. 


Quartic, 6, 28, 35, 85 Simple group, 37, 39, 69, 98 
Quintic, 91 Solution by radicals, 40, 64, 70, 77, 
Quotient-group, 68 81, 91, 92 

Subgroup, 17, 22 (note) 
Rationality, 43 Substitution, 10, 87 
Rational function, 45 Symmetric function, 23, 99 
— relation, 53 — group, 15, 37, 40, 55 
Rectangular array, 22 
Reducible, 46, 59 Transform, 33 
Regular, 59, 68, 70, 73 Transitive, 58 
Resolvent, 5, 24, 49, 60, 64 Transposition, 13 
Self-conjugate, 32 Unaltered, 16, 45, 56, 60 


Series of composition, 37, 40, 97 Uniserial, 73 (note) 








it - = r os 





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5 aS Sage 
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is : ‘ J Wa 
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cee: om | wi ] = 


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vietelete-phes 


